Students simplify and compare expressions. They use rational exponents and simplify square roots.
A1.1.1
Compare real number expressions.
Example: Which is larger: 23 or\sqrt{49}.?
A1.1.2
Simplify square roots using factors.
Example: Explain why \sqrt{48}=4 \sqrt{3}.
A1.1.3
Understand and use the distributive, associative, and commutative properties.
Example: Simplify (6x^2 – 5x + 1) – 2(x^2 + 3x – 4) by removing the parentheses and rearranging. Explain why you can carry out each step.
A1.1.4
Use the laws of exponents for rational exponents.
Example: Simplify 25 .
A1.1.5
Use dimensional (unit) analysis to organize conversions and computations.
Example: Convert 5 miles per hour to feet per second: \frac{{5{\rm{ mi}}}}{{1{\rm{ hr}}}} \cdot \frac{{1{\rm{ hr}}}}{{3{\rm{600 sec}}}} \cdot \frac{{5280{\rm{ ft}}}}{{1{\rm{ mi}}}} \approx 7.3 ft/sec .
Standard 2: Linear Equations and Inequalities
Students solve linear equations and inequalities in one variable. They solve word problems that involve linear equations, inequalities, or formulas.
A1.2.1
Solve linear equations.
Example: Solve the equation 7a + 2 = 5a – 3a + 8 .
A1.2.2
Solve equations and formulas for a specified variable.
Example: Solve the equation q = 4p – 11 for p.
A1.2.3
Find solution sets of linear inequalities when possible numbers are given for the variable.
Example: Solve the inequality 6x – 3 > 10 for x in the set {0, 1, 2, 3, 4}.
A1.2.4
Solve linear inequalities using properties of order.
Example: Solve the inequality 8x – 7 ≤ 2x + 5 , explaining each step in your solution.
A1.2.5
Solve combined linear inequalities.
Example: Solve the inequalities -7 < 3x + 5 < 11 .
A1.2.6
Solve word problems that involve linear equations, formulas, and inequalities.
Example: You are selling tickets for a play that cost $3 each. You want to sell at least $50 worth. Write and solve an inequality for the number of tickets you must sell.
Standard 3: Relations and Functions
Students sketch and interpret graphs representing given situations. They understand the concept of a function and analyze the graphs of functions.
A1.3.1
Sketch a reasonable graph for a given relationship.
Example: Sketch a reasonable graph for a person’s height from age 0 to 25.
A1.3.2
Interpret a graph representing a given situation.
Example: Jessica is riding a bicycle. The graph below shows her speed as it relates to the time she has spent riding. Describe what might have happened to account for such a graph.
A1.3.3
Understand the concept of a function, decide if a given relation is a function, and link equations to functions.
Example: Use either paper or a spreadsheet to generate a list of values for x and y in y = x^2 . Based on your data, make a conjecture about whether or not this relation is a function. Explain your reasoning.
A1.3.4
Find the domain and range of a relation.
Example: Based on the list of values from the last example, what are the domain and range of y = x^2 ?
Standard 4: Graphing Linear Equations and Inequalities
Students graph linear equations and inequalities in two variables. They write equations of lines and find and use the slope and y-intercept of lines. They use linear equations to model real data.
A1.4.1
Graph a linear equation.
Example: Graph the equation 3x – y = 2 .
A1.4.2
Find the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line.
Example: Find the slope and y-intercept of the line 4x + 6y = 12 .
A1.4.3
Write the equation of a line in slope-intercept form. Understand how the slope and y-intercept of the graph are related to the equation.
Example: Write the equation of the line 4x + 6y = 12 in slope-intercept form. What is the slope of this line? Explain your answer.
A1.4.4
Write the equation of a line given appropriate information.
Example: Find an equation of the line through the points (1, 4) and (3, 10), then find an equation of the line through the point (1, 4) perpendicular to the first line.
A1.4.5
Write the equation of a line that models a data set and use the equation (or the graph of the equation) to make predictions. Describe the slope of the line in terms of the data, recognizing that the slope is the rate of change.
Example: As your family is traveling along an interstate, you note the distance traveled every 5 minutes. A graph of time and distance shows that the relation is approximately linear. Write the equation of the line that fits your data. Predict the time for a journey of 50 miles. What does the slope represent?
A1.4.6
Graph a linear inequality in two variables.
Example: Draw the graph of the inequality 6x + 8y ≥ 24 on a coordinate plane.
Standard 5: Pairs of Linear Equations and Inequalities
Students solve pairs of linear equations using graphs and using algebra. They solve pairs of linear inequalities using graphs. They solve word problems involving pairs of linear equations.
A1.5.1
Use a graph to estimate the solution of a pair of linear equations in two variables.
Example: Graph the equations 3y – x = 0 and 2x + 4y = 15 to find where the lines intersect.
A1.5.2
Use a graph to find the solution set of a pair of linear inequalities in two variables.
Example: Graph the inequalities y ≤ 4 and x + y ≤ 5 . Shade the region where both inequalities are true.
A1.5.3
Understand and use the substitution method to solve a pair of linear equations in two variables.
Example: Solve the equations y = 2x and 2x + 3y = 12 by substitution.
A1.5.4
Understand and use the addition or subtraction method to solve a pair of linear equations in two variables.
Example: Use subtraction to solve the equations: 3x + 4y = 11 and 3x + 2y = 7 .
A1.5.5
Understand and use multiplication with the addition or subtraction method to solve a pair of linear equations in two variables.
Example: Use multiplication with the subtraction method to solve the equations: x + 4y = 16 and 3x + 2y = -3 .
A1.5.6
Use pairs of linear equations to solve word problems.
Example: The income a company makes from a certain product can be represented by the equation y = 10.5x and the expenses for that product can be represented by the equation y = 5.25x + 10,000 , where x is the amount of the product sold and y is the number of dollars. How much of the product must be sold for the company to reach the break-even point?
Standard 6: Polynomials
Students add, subtract, multiply, and divide polynomials. They factor quadratics.
Divide polynomials by monomials.
Example: Divide 4x^3y^2 + 8xy^4 – 6x^2y^5 by 2xy^2 .
A1.6.6
Find a common monomial factor in a polynomial.
Example: Factor 36xy^2 + 18xy^4 – 12x^2y^4 .
A1.6.7
Factor the difference of two squares and other quadratics.
Example: Factor 4x^2 – 25 and 2x^2 – 7x + 3 .
A1.6.8
Understand and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.
Example: A graphing calculator can be used to solve 3x^2 – 5x – 1 = 0 to the nearest tenth. Justify using the x-intercepts of y = 3x^2 – 5x – 1 as the solutions of the equation.
Standard 7: Algebraic Fractions
Students simplify algebraic ratios and solve algebraic proportions.
Solve algebraic proportions.
Example: Create a tutorial to be posted to the school’s Web site to instruct beginning students in the steps involved in solving an algebraic proportion. Use \frac{{x + 5}}{4} = \frac{{3x + 5}}{7} as an example.
Standard 8: Quadratic, Cubic, and Radical Equations
Students graph and solve quadratic and radical equations. They graph cubic equations.
A1.8.1
Graph quadratic, cubic, and radical equations.
Example: Draw the graph of y = x^2 – 3x + 2 . Using a graphing calculator or a spreadsheet (generate a data set), display the graph to check your work.
A1.8.2
Solve quadratic equations by factoring.
Example: Solve the equation x^2 – 3x + 2 = 0 by factoring.
A1.8.3
Solve quadratic equations in which a perfect square equals a constant.
Example: Solve the equation (x – 7)^2 = 64 .
A1.8.4
Complete the square to solve quadratic equations.
Example: Solve the equation x^2 – 7x + 9 = 0 by completing the square.
A1.8.5
Derive the quadratic formula by completing the square.
Example: Prove that the equation ax^2 + bx + c = 0 has solutions x = \frac{{{\rm{ - }}b \pm \sqrt {b^2 - 4ac} }}{{2a}}.
A1.8.6
Solve quadratic equations using the quadratic formula.
Example: Solve the equation x^2 – 7x + 9 = 0 .
A1.8.7
Use quadratic equations to solve word problems.
Example: A ball falls so that its distance above the ground can be modeled by the equation
s = 100 – 16t^2 , where s is the distance above the ground in feet and t is the time in seconds. According to this model, at what time does the ball hit the ground?
A1.8.8
Solve equations that contain radical expressions.
Example: Solve the equation \sqrt {x + 6} = x.
A1.8.9
Use graphing technology to find approximate solutions of quadratic and cubic equations.
Example: Use a graphing calculator to solve 3x^2 – 5x – 1 = 0 to the nearest tenth.
Standard 9: Mathematical Reasoning and Problem Solving
Students use a variety of strategies to solve problems.
A1.9.1
Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards.
Example: Fran has scored 16, 23, and 30 points in her last three games. How many points must she score in the next game so that her four-game average does not fall below 20 points?
A1.9.2
Decide whether a solution is reasonable in the context of the original situation.
Example: John says the answer to the problem in the first example is 10 points. Is his answer reasonable? Why or why not?
Students develop and evaluate mathematical arguments and proofs.
A1.9.3
Use the properties of the real number system and the order of operations to justify the steps of simplifying functions and solving equations.
Example: Given an argument (such as 3x + 7 > 5x + 1 , and therefore -2x > -6 , and therefore x > 3 ), provide a visual presentation of a step-by-step check, highlighting any errors in the argument.
A1.9.4
Understand that the logic of equation solving begins with the assumption that the variable is a number that satisfies the equation and that the steps taken when solving equations create new equations that have, in most cases, the same solution set as the original. Understand that similar logic applies to solving systems of equations simultaneously.
Example: Try “solving” the equations x + 3y = 5 and 5x + 15y = 25 simultaneously. Explain what went wrong.
A1.9.5
Decide whether a given algebraic statement is true always, sometimes, or never (statements involving linear or quadratic expressions, equations, or inequalities).
Example: Is the statement x^2 – 5x + 2 = x^2 + 5x + 2 true for all x, for some x, or for no x? Explain your answer.
A1.9.6
Distinguish between inductive and deductive reasoning, identifying and providing examples of each.
Example: What type of reasoning are you using when you look for a pattern?
A1.9.7
Identify the hypothesis and conclusion in a logical deduction.
Example: What is the hypothesis and conclusion in this argument: If there is a number x such that 2x + 1 = 7 , then x = 3 ?
A1.9.8
Use counterexamples to show that statements are false, recognizing that a single counterexample is sufficient to prove a general statement false.
Example: Use the demonstration-graphing calculator on an overhead projector to produce an example showing that this statement is false: all quadratic equations have two different solutions.
Students find lengths and midpoints of line segments. They describe and use parallel and perpendicular lines. They find slopes and equations of lines.
G.1.1
Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems.
Example: Find the length and midpoint of the line joining the points A (3, 8) and B (9, 0).
G.1.2
Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used.
Example: Construct the perpendicular bisector of a given line segment, justifying each step of the process.
G.1.3
Understand and use the relationships between special pairs of angles formed by parallel lines and transversals.
Example: In the diagram, the lines k and l are parallel. What is the measure of angle x? Explain your answer.
G.1.4
Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines.
Example: Find an equation of a line perpendicular to y = 4x – 2.
Standard 2: Polygons
Students identify and describe polygons and measure interior and exterior angles. They use congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and areas.
G.2.1
Identify and describe convex, concave, and regular polygons.
Example: Draw a regular hexagon. Is it convex or concave? Explain your answer.
G.2.2
Find measures of interior and exterior angles of polygons, justifying the method used.
Example: Calculate the measure of one interior angle of a regular octagon. Explain your method.
G.2.3
Use properties of congruent and similar polygons to solve problems.
Example: Divide a regular hexagon into triangles by joining the center to each vertex. Show that these triangles are all the same size and shape and find the sizes of the interior angles of the hexagon.
G.2.4
Apply transformations (slides, flips, turns, expansions, and contractions) to polygons to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips, and turns are congruent to the original shape.
Example: Use a drawing program to create regular hexagons, regular octagons, and regular pentagons. Under the drawings, describe which of the polygons would be best for tiling a rectangular floor. Explain your reasoning.
G.2.5
Find and use measures of sides, perimeters, and areas of polygons. Relate these measures to each other using formulas.
Example: A rectangle of area 360 square yards is ten times as long as it is wide. Find its length and width.
G.2.6
Use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity.
Example: Is the polygon formed by connecting the points (2, 1), (6, 2), (5, 6), and (1, 5) a square?
Standard 3: Quadrilaterals
Students identify and describe simple quadrilaterals. They use congruence and similarity. They find measures of sides, perimeters, and areas.
G.3.1
Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite.
Example: Use a drawing program to create a square, rectangle, rhombus, parallelogram, trapezoid, and kite. Judge which of the quadrilaterals has perpendicular diagonals and draw those diagonals in the figures. Give a convincing argument that your judgment is correct.
G.3.2
Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas.
Example: Of two similar rectangles, the second has sides three times the length of the first. How many times larger in area is the second rectangle?
G.3.3
Find and use measures of sides, perimeters, and areas of quadrilaterals. Relate these measures to each other using formulas.
Example: A section of roof is a trapezoid with length 4 m at the ridge and 6 m at the gutter. The shortest distance from ridge to gutter is 3 m. Construct a model using a drawing program, showing how to find the area of this section of roof.
G.3.4
Use coordinate geometry to prove properties of quadrilaterals, such as regularity, congruence, and similarity.
Example: Is rectangle ABCD with vertices at (0, 0), (4, 0), (4, 2), (0, 2) congruent to rectangle PQRS with vertices at (-2, -1), (2, -1), (2, 1), (-2, 1)?
Standard 4: Triangles
Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They apply inequality theorems.
G.4.1
Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.
Example: Use a drawing program to create examples of right, acute, obtuse, scalene, isosceles, equilateral, and equiangular triangles. Identify and describe the attributes of each triangle.
G.4.2
Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors.
Example: Draw several triangles. Construct their angle bisectors. What do you notice?
G.4.3
Construct triangles congruent to given triangles.
Example: Construct a triangle given the lengths of two sides and the measure of the angle between the two sides.
G.4.4
Use properties of congruent and similar triangles to solve problems involving lengths and areas.
Example: Of two similar triangles, the second has sides half the length of the first. The area of the first triangle is 20 cm2. What is the area of the second?
G.4.5
Prove and apply theorems involving segments divided proportionally.
Example: In triangle ABC, is parallel to . What is the length of ?
G.4.6
Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.
Example: In the last example, prove that triangles ABC and APQ are similar.
G.4.7
Find and use measures of sides, perimeters, and areas of triangles. Relate these measures to each other using formulas.
Example: The gable end of a house is a triangle 20 feet long and 13 feet high. Find its area.
G.4.8
Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and the hinge theorem.
Example: Can you draw a triangle with sides of length 7 cm, 4 cm, and 15 cm?
G.4.9
Use coordinate geometry to prove properties of triangles such as regularity, congruence, and similarity.
Example: Draw a triangle with vertices at (1, 3), (2, 5), and (6, 1). Draw another triangle with vertices at (-3, -1), (-2, 1), and (2, -3). Are these triangles the same shape and size?
Standard 5: Right Triangles
Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the trigonometric relations sine, cosine, and tangent.
G.5.1
Prove and use the Pythagorean Theorem.
Example: On each side of a right triangle, draw a square with that side of the triangle as one side of the square. Find the areas of the three squares. What relationship is there between the areas?
G.5.2
State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.
Example: In triangle ABC with right angle at C, draw the altitude from C to . Name all similar triangles in the diagram. Use these similar triangles to prove the Pythagorean Theorem.
G.5.3
Use special right triangles (30° - 60° and 45° - 45°) to solve problems.
Example: An isosceles right triangle has one short side of 6 cm. Find the lengths of the other two sides.
G.5.4
Define and use the trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles.
Example: In triangle ABC, tan A = . Find sin A and cot A.
G.5.5
Know and use the relationship sin2 x + cos2 x = 1.
Example: Show that, in a right triangle, sin2 x + cos2 x = 1 is an example of the Pythagorean Theorem.
G.5.6
Solve word problems involving right triangles.
Example: The force of gravity pulling an object down a hill is its weight multiplied by the sine of the angle of elevation of the hill. What is the force on a 3,000-pound car on a hill with a 1 in 5 grade? (A grade of 1 in 5 means that the hill rises one unit for every five horizontal units.)
Standard 6: Circles
Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and areas. They prove theorems about circles. They find equations of circles.
G.6.1
Find the center of a given circle. Construct the circle that passes through three given points not on a line.
Example: Given a circle, find its center by drawing the perpendicular bisectors of two chords.
G.6.2
Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent.
Example: What is the angle between a tangent to a circle and the radius at the point where the tangent meets the circle?
G.6.3
Prove theorems related to circles.
Example: Prove that an inscribed angle in a circle is half the measure of the central angle with the same arc.
G.6.4
Construct tangents to circles and circumscribe and inscribe circles.
Example: Draw an acute triangle and construct the circumscribed circle.
G.6.5
Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).
Example: Find the measure of angle ABC in the diagram below.
G.6.6
Define and identify congruent and concentric circles.
Example: Are circles with the same center always the same shape? Are they always the same size?
G.6.7
Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems.
Example: Which will give you more: three 6-inch pizzas or two 8-inch pizzas? Explain your answer.
G.6.8
Find the equation of a circle in the coordinate plane in terms of its center and radius.
Example: Find the equation of the circle with radius 10 and center (6, -3).
Standard 7: Polyhedra and Other Solids
Students describe and make polyhedra and other solids. They describe relationships and symmetries, and use congruence and similarity.
G.7.1
Describe and make regular and nonregular polyhedra.
Example: Is a cube a regular polyhedron? Explain why or why not.
G.7.2
Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron.
Example: Make a net for a tetrahedron out of poster board and fold it up to make the tetrahedron.
G.7.3
Describe relationships between the faces, edges, and vertices of polyhedra.
Example: Count the sides, edges, and corners of a square-based pyramid. How are these numbers related?
G.7.4
Describe symmetries of geometric solids.
Example: Describe the rotation and reflection symmetries of a square-based pyramid.
G.7.5
Describe sets of points on spheres: chords, tangents, and great circles.
Example: On Earth, is the equator a great circle?
G.7.6
Identify and know properties of congruent and similar solids.
Example: Explain how the surface area and volume of similar cylinders are related.
G.7.7
Find and use measures of sides, volumes of solids, and surface areas of solids. Relate these measures to each other using formulas.
Example: An ice cube is dropped into a glass that is roughly a right cylinder with a 6 cm diameter. The water level rises 1 mm. What is the volume of the ice cube?
Standard 8: Mathematical Reasoning and Problem Solving
Students use a variety of strategies to solve problems.
G.8.1
Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards.
Example: How far does the tip of the minute hand of a clock move in 20 minutes if the tip is 4 inches from the center of the clock?
G.8.2
Decide whether a solution is reasonable in the context of the original situation.
Example: John says the answer to the problem in the first example is 12 inches. Is his answer reasonable? Why or why not?
Students develop and evaluate mathematical arguments and proofs.
G.8.3
Make conjectures about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture.
Example: Calculate the ratios of side lengths in several different-sized triangles with angles of 90°, 50°, and 40°. What do you notice about the ratios? How might you prove that your observation is true (or show that it is false)?
G.8.4
Write and interpret statements of the form “if – then” and “if and only if.”
Example: Decide whether this statement is true: “If today is Sunday, then we have school tomorrow.”
G.8.5
State, use, and examine the validity of the converse, inverse, and contrapositive of “if – then” statements.
Example: In the last example, write the converse of the statement.
G.8.6
Identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive proofs.
Example: Do you prove axioms from theorems or theorems from axioms?
G.8.7
Construct logical arguments, judge their validity, and give counterexamples to disprove statements.
Example: Find an example to show that triangles with two sides and one angle equal are not necessarily congruent.
G.8.8
Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs.
Example: In triangle LMN, LM = LN. Prove that LMN LNM.
G.8.9
Perform basic constructions, describing and justifying the procedures used. Distinguish between constructing and drawing geometric figures.
Example: Construct a line parallel to a given line through a given point not on the line, explaining and justifying each step.
Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
A2.1.1
Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
Example: Draw the graphs of the functions y = x^4 – x^2, y = \frac{7}{x-2}, and y = \sqrt{x+2}.
A2.1.2
Use function notation. Add, subtract, multiply, and divide pairs of functions.
Example: Let f(x) = 7x + 2 and g(x) = x^2. Find the value of f(x) \cdot g(x).
A2.1.3
Understand composition of functions and combine functions by composition.
Example: Let f(x) = x^3 and g(x) = x – 2. Find f(g(x)).
A2.1.4
Graph relations and functions with and without graphing technology.
Example: Draw the graph of y = x^3 – 3x^2 – x + 3.
A2.1.5
Find the zeros of a function.
Example: In the last example, find the zeros of the function; i.e., find x when y = 0.
A2.1.6
Solve an inequality by examining the graph.
Example: Find the solution for x^3 – 3x^2 – x + 3<0 by graphing y = x^3 – 3x^2 – x + 3.
A2.1.7
Graph functions defined piece-wise.
Example: Sketch the graph of f(x) = \left\{ \begin{array}{l} x + 2{\rm{ for }}x \ge 0 \\ {\rm{ - }}x^2 {\rm{ for }}x > 0 \\ \end{array} \right. .
A2.1.8
Interpret given situations as functions in graphs, formulas, and words.
Example: You and your parents are going to Boston and want to rent a car at Logan International Airport on a Monday morning and drop the car off in downtown Providence, R.I., on the following Wednesday. Find the rates from two national car companies and plot the costs on a graph. Decide which company offers the best deal. Explain your answer.
Standard 2: Linear and Absolute Value Equations and Inequalities
Students solve systems of linear equations and inequalities and use them to solve word problems. They model data with linear equations.
A2.2.1
Graph absolute value equations and inequalities.
Example: Draw the graph of y = 2x – 5 and use that graph to draw the graph of y = |2x – 5|.
A2.2.2
Use substitution, elimination, and matrices to solve systems of two or three linear equations in two or three variables.
Example: Solve the system of equations: x – 2y + 3z = 5, x + 3z = 11, 5y – 6z = 9.
A2.2.3
Use systems of linear equations and inequalities to solve word problems.
Example: Each week you can work no more than 20 hours all together at the local bookstore and the drugstore. You prefer the bookstore and want to work at least 10 more hours there than at the drugstore. Draw a graph to show the possible combinations of hours that you could work.
A2.2.4
Find a linear equation that models a data set using the median fit method and use the model to make predictions.
Example: You light a candle and record its height in centimeters every minute.
The results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.5), (3, 14.8), (4, 13.2),
(5, 11.5), (6, 10.0), (7, 8.2), (9, 4.9), and (10, 3.1). Find the median fit line to express
the candle’s height as a function of the time and state the meaning of the slope in terms
of the burning candle.
Standard 3: Quadratic Equations and Functions
Students solve quadratic equations, including the use of complex numbers. They interpret maximum and minimum values of quadratic functions. They solve equations that contain square roots.
A2.3.1
Define complex numbers and perform basic operations with them.
Example: Multiply 7 – 4i and 10 + 6i.
A2.3.2
Understand how real and complex numbers are related, including plotting complex numbers as points in the plane.
Example: Plot the points corresponding to 3 – 2i and 1 + 4i. Add these complex numbers and plot the result. How is this point related to the other two?
A2.3.3
Solve quadratic equations in the complex number system.
Example: Solve x^2 – 2x + 5 = 0 over the complex numbers.
A2.3.4
Graph quadratic functions. Apply transformations to quadratic functions. Find and interpret the zeros and maximum or minimum value of quadratic functions.
Example: Find the zeros for y = x^2 – 4. If y = x^2 – 4 has a maximum or minimum value, give the ordered pair corresponding to the maximum or minimum point.
A2.3.5
Solve word problems using quadratic equations.
Example: You have 100 feet of fencing to make three sides of a rectangular area using an existing straight fence as the fourth side. Construct a formula in a spreadsheet to determine the area you can enclose and use the spreadsheet to make a conjecture about the maximum area possible. Prove (or disprove) your conjecture by solving an appropriate quadratic equation.
A2.3.6
Solve equations that contain radical expressions.
Example: Solve the equation \sqrt{x+9} = 9 – \sqrt{x} .
A2.3.7
Solve pairs of equations, one quadratic and one linear or both quadratic.
Example: Solve the system of equations y = x^2 – 5x + 1, x + y + 2 = 0.
Standard 4: Conic Sections
Students write equations of conic sections and draw their graphs.
A2.4.1
Write the equations of conic sections (circle, ellipse, parabola, and hyperbola).
Example: Write an equation for a parabola with focus (2, 3) and directrix y = 1.
A2.4.2
Graph conic sections.
Example: Graph the circle described by the equation (x + 4)^2 + (y – 1)^2 = 9.
Standard 5: Polynomials
Students use the binomial theorem, divide and factor polynomials, and solve polynomial equations.
A2.5.1
Understand the binomial theorem and use it to expand binomial expressions raised to positive integer powers.
Example: Expand (x + 2)^4.
A2.5.2
Divide polynomials by others of lower degree.
Example: Divide 2x^3 – 3x^2 + x – 6 by x^2 + 2.
A2.5.3
Factor polynomials completely and solve polynomial equations by factoring.
Example: Solve x^3 + 27 = 0 by factoring.
A2.5.4
Use graphing technology to find approximate solutions for polynomial equations.
Example: Approximate the solution(s) of x^4 – 3x^3 + 2x – 7 = 0 to the nearest tenth.
A2.5.5
Use polynomial equations to solve word problems.
Example: You want to make an open-top box with a volume of 500 square inches from a piece of cardboard that is 25 inches by 15 inches by cutting squares from the corners and folding up the sides. Find the possible dimensions of the box.
A2.5.6
Write a polynomial equation given its solutions.
Example: Write an equation that has solutions x = 2, x = 5i and x = -5i.
A2.5.7
Understand and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.
Example: Solve the equation x^4 + x^3 – 7x^2 – x + 6 = 0, given that x – 2 and x + 3 are factors of x^4 + x^3 – 7x^2 – x + 6.
Standard 6: Algebraic Fractions
Students use negative and fractional exponents. They simplify algebraic fractions and solve equations involving algebraic fractions. They solve problems of direct, inverse, and joint variation.
A2.6.1
Understand and use negative and fractional exponents.
Example: Simplify (2a^{-2}b^3)^4 (4a^3b^{-1})^{-2} .
Solve word problems involving fractional equations.
Example: Two students, working independently, can complete a particular job in 20 minutes and 30 minutes, respectively. How long will it take to complete the job if they work together?
A2.6.6
Solve problems of direct, inverse, and joint variation.
Example: One day your drive to work takes 10 minutes and you average 30 mph. The next day the drive takes 15 minutes. What is your average speed that day?
Standard 7: Logarithmic and Exponential Functions
Students graph exponential functions and relate them to logarithms. They solve logarithmic and exponential equations and inequalities. They solve word problems using exponential functions.
A2.7.1
Graph exponential functions.
Example: Draw the graphs of the functions y = 2^x and y = 2^{-x} .
A2.7.2
Prove simple laws of logarithms.
Example: Use the fact that a^x \cdot a^y = a^{x + y} to show that log_a (pq) = log_a p + log_a q.
A2.7.3
Understand and use the inverse relationship between exponents and logarithms.
Example: Find the value of log_10(10^7).
A2.7.4
Solve logarithmic and exponential equations and inequalities.
Example: Solve the equation log_2 x = 5.
A2.7.5
Use the definition of logarithms to convert logarithms from one base to another.
Example: Write log_10 75 as a logarithm to base 2.
A2.7.6
Use the properties of logarithms to simplify logarithmic expressions and to find their approximate values.
Example: Simplify log_3 81.
A2.7.7
Use calculators to find decimal approximations of natural and common logarithmic numeric expressions.
Example: Find a decimal approximation for ln 500.
A2.7.8
Solve word problems involving applications of exponential functions to growth and decay.
Example: The population of a certain country can be modeled by the equation P(t) = 50e0.02t, where P is the population in millions and t is the number of years after 1900. Find when the population is 100 million, 200 million, and 400 million. What do you notice about these time periods?
Standard 8: Sequences and Series
Students define and use arithmetic and geometric sequences and series.
A2.8.1
Define arithmetic and geometric sequences and series.
Example: What type of sequence is 10, 100, 1,000, 10,000, …?
A2.8.2
Find specified terms of arithmetic and geometric sequences.
Example: Find the tenth term of the arithmetic sequence 3, 7, 11, 15, …
A2.8.3
Find partial sums of arithmetic and geometric series.
Example: In the last example, find the sum of the first 10 terms.
A2.8.4
Solve word problems involving applications of sequences and series.
Example: You have on a Petri dish 1 square millimeter of a mold that doubles in size each day. What area will it cover after a month?
Standard 9: Counting Principles and Probability
Students use fundamental counting principles to compute combinations, permutations, and probabilities.
A2.9.1
Understand and apply counting principles to compute combinations and permutations.
Example: There are 5 students who work in a bookshop. If the bookshop needs 3 people to operate, how many days straight could the bookstore operate without the same group of students working twice?
A2.9.2
Use the basic counting principle, combinations, and permutations to compute probabilities.
Example: You are on a chess team made up of 15 players. What is the probability that you will be chosen if a 3-person team is selected at random?
Standard 10: Mathematical Reasoning and Problem Solving
Students use a variety of strategies to solve problems.
A2.10.1
Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards.
Example: The swimming pool at Roanoke Park is 24 feet long and 18 feet wide. The park district has determined that they have enough money to put a walkway of uniform width, with a maximum area of 288 square feet, around the pool. How could you find the maximum width of a new walkway?
A2.10.2
Decide whether a solution is reasonable in the context of the original situation.
Example: John says the answer to the problem in the first example is 20 feet. Is that reasonable?
Students develop and evaluate mathematical arguments and proofs.
A2.10.3
Decide if a given algebraic statement is true always, sometimes, or never (statements involving rational or radical expressions or logarithmic or exponential functions).
Example: Is the statement (a^x)^y = a^{xy} true for all x, for some x, or for no x?
A2.10.4
Use the properties of number systems and the order of operations to justify the steps of simplifying functions and solving equations.
Example: Simplify 2(x^3– 3x^2 + x – 6) – (x – 3)(x + 4), explaining why you can take each step.
A2.10.5
Understand that the logic of equation solving begins with the assumption that the variable is a number that satisfies the equation and that the steps taken when solving equations create new equations that have, in most cases, the same solution set as the original. Understand that similar logic applies to solving systems of equations simultaneously.
Example: A student solving the equation \sqrt{x+6} = x comes up with the solution set {-2, 3}. Explain why {-2, 3} is not the solution set to this equation, and why the “check” step is essential in solving the equation.
A2.10.6
Use counterexamples to show that statements are false.
Example: Show by an example that this statement is false: The product of two complex numbers is never a real number.
Students use polynomial, rational, and algebraic functions to write functions and draw graphs to solve word problems, to find composite and inverse functions, and to analyze functions and graphs. They analyze and graph circles, ellipses, parabolas, and hyperbolas.
PC.1.1
Recognize and graph various types of functions, including polynomial, rational, algebraic, and absolute value functions. Use paper and pencil methods and graphing calculators.
Example: Draw the graphs of the functions y = x^5 – 2x^3 – 5x^2, y =\frac{2x-1}{3x+2}, and y =\sqrt{(x+2)(x-5)}.
PC.1.2
Find domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions. Use paper and pencil methods and graphing calculators.
Example: Let R(x) =\frac{1}{\sqrt{x-2}}. Find the domain of R(x) — i.e., the values of x for which R(x) is defined. Also find the range, zeros, and asymptotes of R(x).
PC.1.3
Model and solve word problems using functions and equations.
Example: You are on the committee for planning the prom and need to decide what to charge for tickets. Last year you charged $5.00 and 400 people bought tickets. Earlier experiences suggest that for every 10¢ decrease in price you will sell 50 extra tickets. Use a spreadsheet and write a function to show how the amount of money in ticket sales depends on the number of 10¢ decreases in price. Construct a graph that shows the price and gross receipts. What is the optimum price you should set for the tickets?
PC.1.4
Define, find, and check inverse functions.
Example: Find the inverse function of h(x) = (x – 2)^3.
PC.1.5
Describe the symmetry of the graph of a function.
Example: Describe the symmetries of the functions x, x^2, x^3, and x^4.
PC.1.6
Decide if functions are even or odd.
Example: Is the function tan x even, odd, or neither? Explain your answer.
PC.1.7
Apply transformations to functions.
Example: Explain how you can obtain the graph of g(x) = -|2(x + 3)^2 – 2| from the graph of f(x) = x^2.
PC.1.8
Understand curves defined parametrically and draw their graphs.
Example: Draw the graph of the function y = f(x), where x = 3t + 1 and y = 2t^2 – 5 for a parameter t.
PC.1.9
Compare relative magnitudes of functions and their rates of change.
Example: Contrast the growth of y = x^2 and y = 2^x.
PC.1.10
Write the equations of conic sections in standard form (completing the square and using translations as necessary), in order to find the type of conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.).
Example: Write the equation x^2 + y^2 – 10x – 6y – 25 = 0 in standard form. Decide what kind of conic it is and find its foci, asymptotes, and eccentricity as appropriate.
Standard 2: Logarithmic and Exponential Functions
Students solve word problems involving logarithmic and exponential functions. They draw and analyze graphs and find inverse functions.
PC.2.1
Solve word problems involving applications of logarithmic and exponential functions.
Example: The amount A gm of a radioactive element after t years is given by the formula
A(t) = 100e^{-0.02t} . Find t when the amount is 50 gm, 25 gm, and 12.5 gm. What do you notice about these time periods?
PC.2.2
Find the domain, range, intercepts, and asymptotes of logarithmic and exponential functions.
Example: For the function L(x) = log_{10} (x – 4), find its domain, range, x-intercept, and asymptote.
PC.2.3
Draw and analyze graphs of logarithmic and exponential functions.
Example: In the last example, draw the graph of L(x).
PC.2.4
Define, find, and check inverse functions of logarithmic and exponential functions.
Example: Find the inverse of f(x) = 3e^{2x}.
Standard 3: Trigonometry in Triangles
Students define trigonometric functions using right triangles. They solve word problems and apply the laws of sines and cosines.
PC.3.1
Solve word problems involving right and oblique triangles.
Example: You want to find the width of a river that you cannot cross. You decide to use a tall tree on the other bank as a landmark. From a position directly opposite the tree, you measure 50 m along the bank. From that point, the tree is in a direction at 37º to your 50 m line. How wide is the river?
PC.3.2
Apply the laws of sines and cosines to solving problems.
Example: You want to fix the location of a mountain by taking measurements from two positions 3 miles apart. From the first position, the angle between the mountain and the second position is 78º. From the second position, the angle between the mountain and the first position is 53º. How far is the mountain from each position?
PC.3.3
Find the area of a triangle given two sides and the angle between them.
Example: Calculate the area of a triangle with sides of length 8 cm and 6 cm enclosing an angle of 60º.
Standard 4: Trigonometric Functions
Students define trigonometric functions using the unit circle and use degrees and radians. They draw and analyze graphs, find inverse functions, and solve word problems.
PC.4.1
Define sine and cosine using the unit circle.
Example: Find the acute angle A for which sin 150º = sin A.
PC.4.2
Convert between degree and radian measures.
Example: Convert 90º, 45º, and 30º to radians.
PC.4.3
Learn exact sine, cosine, and tangent values for 0, π/2, π /3, π/4, π/6, and multiples of π. Use those values to find other trigonometric values.
Example: Find the values of cos(π/2), tan(3π/4), csc(2π/3), sin^{-1} \frac{-\sqrt{3}}{2}, and sin(3π).
PC.4.4
Solve word problems involving applications of trigonometric functions.
Example: In Indiana, the day length in hours varies through the year in a sine wave. The longest day of 14 hours is on Day 175 and the shortest day of 10 hours is on Day 355. Sketch a graph of this function and find its formula. Which other day has the same length as July 4?
PC.4.5
Define and graph trigonometric functions (i.e., sine, cosine, tangent, cotangent, secant, cosecant).
Example: Graph y = sin x and y = cos x, and compare their graphs.
PC.4.6
Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions.
Example: Find the asymptotes of tan x and find its domain.
PC.4.7
Draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift.
Example: Draw the graph of y = 5 + sin (x –\frac{\pi}{3}).
Find values of trigonometric and inverse trigonometric functions.
Example: Find the values of sin(π/2) and tan^{-1} \sqrt{3} .
PC.4.10
Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.
Example: Use a right triangle to show that the slope of a line at 135º to the x-axis is -1.
PC.4.11
Make connections between right triangle ratios, trigonometric functions, and circular functions.
Example: Angle A is a 60º angle of a right triangle with a hypotenuse of length 14 and a shortest side of length 7. Find the exact sine, cosine, and tangent of angle A. Find the real numbers x, 0 < x < 2π, with exactly the same sine, cosine, and tangent values.
Standard 5: Trigonometric Identities and Equations
Students prove trigonometric identities, solve trigonometric equations, and solve word problems.
PC.5.1
Know the basic trigonometric identity cos^2x + sin^2x = 1 and prove that it is equivalent to the Pythagorean Theorem.
Example: Use a right triangle to show that cos^2x + sin^2x = 1.
PC.5.2
Use basic trigonometric identities to verify other identities and simplify expressions.
Example: Show that \frac{tan^2x}{1+tan^2x}=sin^2 x.
PC.5.3
Understand and use the addition formulas for sines, cosines, and tangents.
Example: Prove that sin (A + B) = sin A cos B + cos A sin B and use it to find a formula for sin 2x.
PC.5.4
Understand and use the half-angle and double-angle formulas for sines, cosines, and tangents.
Example: Prove that cos^2x = \frac{1}{2} + \frac{1}{2} cos 2x.
PC.5.5
Solve trigonometric equations.
Example: Solve 3 \, sin 2x = 1 for x between 0 and 2π.
PC.5.6
Solve word problems involving applications of trigonometric equations.
Example: In the example about day length in Standard 4, for how long in winter is there less than 11 hours of daylight?
Standard 6: Polar Coordinates and Complex Numbers
Students define polar coordinates and complex numbers and understand their connection with trigonometric functions.
PC.6.1
Define polar coordinates and relate polar coordinates to Cartesian coordinates.
Example: Convert the polar coordinates (2, π/3) to (x, y) form.
PC.6.2
Represent equations given in rectangular coordinates in terms of polar coordinates.
Example: Represent the equation x^2 + y^2 = 4 in terms of polar coordinates.
PC.6.3
Graph equations in the polar coordinate plane.
Example: Graph y = 1 – cos \theta .
PC.6.4
Define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form.
Example: Write 3 + 3i and 2 – 4i in trigonometric form and then multiply the results.
PC.6.5
State, prove, and use De Moivre’s Theorem.
Example: Simplify (1 – i)^{23} .
Standard 7: Sequences and Series
Students define and use arithmetic and geometric sequences and series, understand the concept of a limit, and solve word problems.
PC.7.1
Understand and use summation notation.
Example: Write the terms of \sum\limits_1^5 {n^2 } .
PC.7.2
Find sums of infinite geometric series.
Example: Find the sum of 1 + 1/2 + 1/4 + 1/8 + 1/16 + …
PC.7.3
Prove and use the sum formulas for arithmetic series and for finite and infinite geometric series.
Example: Prove that a + ar + ar^2 + ar^3 + ar^4 + … = \frac{a}{/ (1 – r)} .
PC.7.4
Use recursion to describe a sequence.
Example: Write the first five terms of the Fibonacci sequence with a_1 = 1, a_2 = 1, and a_n = a_{n-1} + a_{n-2} for n \ge 3 .
PC.7.5
Understand and use the concept of limit of a sequence or function as the independent variable approaches infinity or a number. Decide whether simple sequences converge or diverge.
Example: Find the limit as n \to ∞ of the sequence \frac{{2n - 1}}{{3n + 2}} and \mathop {\lim }\limits_{x \to 5} \frac{{x^2 - 5^2 }}{{x - 5}}.
PC.7.6
Solve word problems involving applications of sequences and series.
Example: You put $100 in your bank account today, and then each day put half the amount of the previous day (always rounding to the nearest cent). Will you ever have $250 in your account?
Standard 8: Data Analysis
Students model data with linear and nonlinear functions.
PC.8.1
Find linear models using the median fit and least squares regression methods. Decide which model gives a better fit.
Example: Measure the wrist and neck size of each person in your class and make a scatterplot. Find the median fit line and the least squares regression line. Which line is a better fit? Explain your reasoning.
PC.8.2
Calculate and interpret the correlation coefficient. Use the correlation coefficient and residuals to evaluate a “best-fit” line.
Example: Calculate and interpret the correlation coefficient for the linear regression model in the last example. Graph the residuals and evaluate the fit of the linear equation.
PC.8.3
Find a quadratic, exponential, logarithmic, power, or sinusoidal function to model a data set and explain the parameters of the model.
Example: Drop a ball and record the height of each bounce. Make a graph of the height (vertical axis) versus the bounce number (horizontal axis). Find an exponential function of the form y = a \cdot b^x that fits the data and explain the implications of the parameters a and b in this experiment.
Standard 9: Mathematical Reasoning and Problem Solving
Students use a variety of strategies to solve problems.
PC.9.1
Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, examining simpler problems, and working backwards.
Example: The half-life of carbon-14 is 5,730 years. The original concentration of carbon-14 in a living organism was 500 grams. How might you find the age of a fossil of that living organism with a carbon-14 concentration of 140 grams?
PC.9.2
Decide whether a solution is reasonable in the context of the original situation.
Example: John says the answer to the problem in the first example is about 10,000 years. Is his answer reasonable? Why or why not?
Students develop and evaluate mathematical arguments and proofs.
PC.9.3
Decide if a given algebraic statement is true always, sometimes, or never (statements involving rational or radical expressions, trigonometric, logarithmic or exponential functions).
Example: Is the statement sin 2x = 2 sin x cos x true for all x, for some x or for no x? Explain your answer.
PC.9.4
Use the properties of number systems and order of operations to justify the steps of simplifying functions and solving equations.
Example: Simplify \left( {\frac{5}{{x - 2}}{\rm{ }} + {\rm{ }}\frac{2}{{x + 3}}} \right) \div \left( {\frac{1}{{x + 3}}{\rm{ }} + {\rm{ }}\frac{7}{{x - 2}}} \right), explaining why you can take each step.
PC.9.5
Understand that the logic of equation solving begins with the assumption that the variable is a number that satisfies the equation, and that the steps taken when solving equations create new equations that have, in most cases, the same solution set as the original. Understand that similar logic applies to solving systems of equations simultaneously.
Example: A student solving the equation x + \sqrt{x} – 30 = 0 comes up with the solution set {25, 36}. Explain why {25, 36} is not the solution set to this equation, and why the “check” step is essential in solving the equation.
PC.9.6
Define and use the mathematical induction method of proof.
Example: Prove De Moivre’s Theorem using mathematical induction
Students understand the concept of limit, find limits of functions at points and at infinity, decide if a function is continuous, and use continuity theorems.
C.1.1
Understand the concept of limit and estimate limits from graphs and tables of values.
Example: Estimate \mathop {\lim }\limits_{x \to 2} \frac{{x^2 + 2x - 8}}{{x - 2}} by calculating the function’s values for x = 2.1, 2.01, 2.001 and for x = 1.9, 1.99, 1.999.
Find limits of sums, differences, products, and quotients.
Example: Find \mathop {\lim }\limits_{x \to \pi} (sin x \cdot cos x+tan x) .
C.1.4
Find limits of rational functions that are undefined at a point.
Example: Find \mathop {\lim }\limits_{x \to 2} \frac{x^2 + 2x - 8}{x - 2} by factoring and canceling.
Decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior.
Example: Find \mathop {\lim }\limits_{x \to 0 } \frac{1}{x^2} .
C.1.8
Find special limits such as \mathop {\lim }\limits_{x \to 0 } \frac{x}{sin x} .
Example: Use a diagram to show that the limit above is equal to 1.
C.1.9
Understand continuity in terms of limits.
Example: Show that f(x) = 3x + 1 is continuous at x = 2 by finding \mathop {\lim }\limits_{x \to 2} (3x+1) and comparing it with f(2) .
C.1.10
Decide if a function is continuous at a point.
Example: Show that f(x) = \frac{x^2 + 2x - 8}{x - 2} is continuous at x = 2 , provided that you define f(2) = 6 .
C.1.11
Find the types of discontinuities of a function.
Example: What types of discontinuities has h(x) = \frac{x^2 -5x+6}{x^2 - 4} ? Explain your answer.
C.1.12
Understand and use the Intermediate Value Theorem on a function over a closed interval.
Example: Show that g(x) = 3 – x^2 has a zero between x = 1 and x = 2 , because it is continuous.
C.1.13
Understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval.
Example: Decide if t(x) = tan x has a maximum value over the interval \left[ { - \frac{\pi }{4},\frac{\pi }{4}} \right] . What about the interval \left[ - \pi,\pi \right] ? Explain your answers.
Standard 2: Differential Calculus
Students find derivatives of algebraic, trigonometric, logarithmic, and exponential functions. They find derivatives of sums, products, and quotients, and composite and inverse functions. They find derivatives of higher order and use logarithmic differentiation and the Mean Value Theorem.
C.2.1
Understand the concept of derivative geometrically, numerically, and analytically, and interpret the derivative as a rate of change.
Example: Find the derivative of f(x) =x^2 at x = 5 by calculating values of \frac{{x^2 - 5^2 }}{{x - 5}} for x near 5. Use a diagram to explain what you are doing and what the result means.
C.2.2
State, understand, and apply the definition of derivative.
Example: Find \mathop {\lim }\limits_{x \to 5} \frac{{x^2 - 5^2 }}{{x - 5}}. What does the result tell you?
C.2.3
Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.
Example: Find \frac{dy }{dx} for the function y = x^5 .
C.2.4
Find the derivatives of sums, products, and quotients.
Example: Find the derivative of x cos x .
C.2.5
Find the derivatives of composite functions, using the chain rule.
Example: Find f'(x) for f(x) = (x^2 + 2)^4 .
C.2.6
Find the derivatives of implicitly-defined functions.
Example: For xy – x^2y^2 = 5 , find \frac{dy }{dx} at the point (2, 3).
C.2.7
Find derivatives of inverse functions.
Example: Let f(x)=2x^3 and g=f^{-1}. Find g'(2) .
C.2.8
Find second derivatives and derivatives of higher order.
Example: Find the second derivative of e^{5x}.
C.2.9
Find derivatives using logarithmic differentiation.
Example: Find \frac{dy }{dx} for y=\sqrt{(x+3)^3(x-7)} .
C.2.10
Understand and use the relationship between differentiability and continuity.
Example: Is f(x)=\frac{1}{x} continuous at x = 0 ? Is f(x) differentiable at x = 0 ? Explain your answers.
C.2.11
Understand and apply the Mean Value Theorem.
Example: For f(x)=\sqrt{x} on the interval [1, 9] , find the value of c such that \frac{{f(9) - f(1)}}{{9 - 1}} = f'(c) .
Standard 3: Applications of Derivatives
Students find slopes and tangents, maximum and minimum points, and points of inflection. They solve optimization problems and find rates of change.
C.3.1
Find the slope of a curve at a point, including points at which there are vertical tangents and no tangents.
Example: Find the slope of the tangent to y =x^3 at the point (2, 8).
C.3.2
Find a tangent line to a curve at a point and a local linear approximation.
Example: In the last example, find an equation of the tangent at (2, 8).
C.3.3
Decide where functions are decreasing and increasing. Understand the relationship between the increasing and decreasing behavior of f and the sign of f' .
Example: Use values of the derivative to find where f(x) = x^3– 3x is decreasing.
C.3.4
Find local and absolute maximum and minimum points.
Example: In the last example, find the local maximum and minimum points of f(x).
C.3.5
Analyze curves, including the notions of monotonicity and concavity.
Example: In the last example, for which values of x is f(x) decreasing and for which values of x is f(x) concave upward?
C.3.6
Find points of inflection of functions. Understand the relationship between the concavity of f and the sign of f" . Understand points of inflection as places where concavity changes.
Example: In the last example, find the points of inflection of f(x) and where f(x) is concave upward and concave downward.
C.3.7
Use first and second derivatives to help sketch graphs. Compare the corresponding characteristics of the graphs of f , f' , and f" .
Example: Use the last examples to draw the graph of f(x) = x^3– 3x .
C.3.8
Use implicit differentiation to find the derivative of an inverse function.
Example: Let f(x) = 2x^3and g = f^{-1}. Find g'(x) using implicit differentiation.
C.3.9
Solve optimization problems.
Example: You want to enclose a rectangular area of 5,000 m^2. Find the shortest length of fencing you can use.
C.3.10
Find average and instantaneous rates of change. Understand the instantaneous rate of change as the limit of the average rate of change. Interpret a derivative as a rate of change in applications, including velocity, speed, and acceleration.
Example: You are filling a bucket with water and the height H cm of the water after t seconds is given by H(t) = (4t)^{\frac{2}{3}} . How fast is the water rising 30 seconds after you start filling the bucket? Explain your answer.
C.3.11
Find the velocity and acceleration of a particle moving in a straight line.
Example: A bead on a wire moves so that, after t seconds, its distance s cm from the midpoint of the wire is given by s = 5 sin (t – \frac {\pi}{4}). Find its maximum velocity and where along the wire this occurs.
C.3.12
Model rates of change, including related rates problems.
Example: A boat is heading south at 10 mph. Another boat is heading west at 15 mph toward the same point. At these speeds, they will collide. Find the rate that the distance between them is decreasing 1 hour before they collide.
Standard 4: Integral Calculus
Students define integrals using Riemann Sums, use the Fundamental Theorem of Calculus to find integrals, and use basic properties of integrals. They integrate by substitution and find approximate integrals.
C.4.1
Use rectangle approximations to find approximate values of integrals.
Example: Find an approximate value for \int_0^3 {x^2 dx} using 6 rectangles of equal width under the graph of f(x) = x^2.
C.4.2
Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points.
Example: Find the value of the Riemann Sum over the interval (:cell PQA(PSS(0, 3):) using 6 subintervals of equal width for f(x) = x^2 evaluated at the midpoint of each subinterval.
C.4.3
Interpret a definite integral as a limit of Riemann Sums.
Example: Find the values of the Riemann Sums over the interval {0, 3$} using 12, 24, etc., subintervals of equal width for f(x) = x^2 evaluated at the midpoint of each subinterval. Find the limit of the Riemann Sums.
C.4.4
Understand the Fundamental Theorem of Calculus: Interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval, that is \int_a^b {f'(x) dx} .
Example: Explain why \int_4^5 {2x dx}=5^2-4^2 ..
C.4.5
Use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined.
Example: Using antiderivatives, find \int_0^3 {x^2 dx} ..
C.4.6
Understand and use these properties of definite integrals:
If f(x) ≤ g(x) on {a, b$}, then \int_a^b {f(x) dx} ≤ \int_a^b {g(x) dx}.
Example: Find \int_0^3 {5x^2) dx}, given that \int_0^3 {x^2) dx}=9.
C.4.7
Understand and use integration by substitution (or change of variable) to find values of integrals.
Example: Find \int_1^2 {x^2(x^3+1)^4) dx}.
C.4.8
Understand and use Riemann Sums, the Trapezoidal Rule, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
Example: Use the Trapezoidal Rule with 6 subintervals over (:cell PQA(PSS(0, 3):) for f(x) = x^2 to approximate the value of \int_0^3 {x^2) dx}.
Standard 5: Applications of Integration
Students find velocity functions and position functions from their derivatives, solve separable differential equations, and use definite integrals to find areas and volumes.
C.5.1
Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions, and applications to motion along a line.
Example: A bead on a wire moves so that its velocity, after t seconds, is given by
v(t) = 3 cos 3t. Given that it starts 2 cm to the left of the midpoint of the wire, find its position after 5 seconds.
C.5.2
Solve separable differential equations and use them in modeling.
Example: The slope of the tangent to the curve y = f(x) is given by \frac{-x}{y} . Find an equation of the curve y = f(x).
C.5.3
Solve differential equations of the form y' = ky as applied to growth and decay problems.
Example: The amount of a certain radioactive material was 10 kg a year ago. The amount is now 9 kg. When will it be reduced to 1 kg? Explain your answer.
C.5.4
Use definite integrals to find the area between a curve and the x-axis, or between two curves.
Example: Find the area bounded by y = \sqrt{x}, x = 0, and x = 2.
C.5.5
Use definite integrals to find the average value of a function over a closed interval.
Example: Find the average value of y = \sqrt{x} over {0, 2$}.
C.5.6
Use definite integrals to find the volume of a solid with known cross-sectional area.
Example: A cone with its vertex at the origin lies symmetrically along the x-axis. The base of the cone is at x = 5 and the base radius is 7. Use integration to find the volume of the cone.
C.5.7
Apply integration to model and solve problems in physics, biology, economics, etc., using the integral as a rate of change to give accumulated change and using the method of setting up an approximating Riemann Sum and representing its limit as a definite integral.
Example: Find the amount of work done by a variable force.
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