Calculus Summer Math Practice. 1. Find inverse functions Describe in words how you use algebra to determine the inverse function.

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1 1 Calculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School Calculus Summer Math Practice Please see the math department document for instructions on setting up an account, entering the Calculus class, and using the website. If you have any questions, be sure to contact me at I will reply within 48 hours, except for June when I am on the mission trip in Nicaragua. Study Guide Questions NOTE: The last 5 skills listed are only required for students entering AP Calculus AB. However, due to possible changes in schedule during the summer, they will be recommended for all Calculus students. Students in Regular Calculus may ignore those 5 recommendations. 1. Find inverse functions Describe in words how you use algebra to determine the inverse function. Determine the inverse function for the following functions. Show ALL work. f(x) = 2xx+8 5 g(x) = 3xx Transforming functions Match the functions below to the following transformations: shift right; shift left; shift up; shift down; vertical stretch taller; vertical shrink shorter; vertical reflection (upside-down); horizontal reflection (sideways) g(x) = f(x+2) h(x) = f(x) 4 k(x) = f(x-3) q(x) = f(x) + 7 r(x) = 3f(x) p(x) = 1 ff(xx) v(x) = -f(x) w(x) = f(-x) 2

2 2 3. Find composite functions Describe in your own words what the expression f(g(x)) means. Given that f(x) = 3x 1 and g(x) = x 2 + 4, calculate the following. Show ALL work. f(g(4)) g(f(4)) f(g(x)) g(f(x)) 4. Evaluate composite functions: graphs & tables Explain your reasoning as you solve these two problems screenshot from Khan Academy. 5. Positive and negative intervals 6. Increasing and decreasing intervals Given the graph below, identify the positive, negative, increasing, and decreasing intervals. Explain why each interval is described by that word. Use approximations, if necessary. positive intervals: negative intervals: increasing intervals: decreasing intervals:

3 3 7. Evaluate logarithms When you complete this skill, what number is your answer? For example, why is 3 is the answer to the expression log 2 8? Evaluate the logarithms below. Justify that each answer is correct. log 4 64 log log 2 1, Evaluate logarithms (advanced) What exponent is equivalent to the square root function? What does a negative exponent do to an expression? For example, what is 7-2? What is true of any number raised to the power of zero? Evaluate the logarithms below. Justify that each answer is correct. 1 log 16 4 log 5 log log 8 1

4 4 9. Relationship between exponentials & logarithms What is true of the relationship between an exponential expression and a logarithm expression? Given what you said in Skill #1, what does this mean about these functions? Complete the following exercises. Describe your reasoning below. Re-write the expression 3 4 = 81 in logarithm form. Re-write the expression log 2 16 = 4 in exponent form. 10. Use the properties of logarithms List the 5 properties of logarithms below. Describe what they mean in your own words. Write the following logarithms as a single logarithm expression. log(4) + 2log(6) log(10)-log(2) 3log(6)-log(9)

5 5 11. Solve exponential equations using logarithms: base-10 and base-e When you see log with no base written, what is the base understood to be? When you see natural log ( ln ), what is the base understood to be? Solve the following equations for x. Show all steps. Include both the exact answer and a decimal approximation to 3 digits tt = 20 2e 7x + 3 = 19

6 6 12. Trig values of special angles On the Unit Circle, the cosine function is the value and the sine function is the value. Complete the Unit Circle below. Give the x and y-values as well as the angle measure in degrees and radians.

7 7 13. Graph sinusoidal functions Given an equation in the form of yy = a sin b(x c) + d, describe what each term means on the graph and how you calculate the following: Amplitude: Period: Midline: Graph the following trig functions. Be sure to label the amplitude, period, and midline. y = 3sin(4x) + 7 y = -4cos (πx) Evaluate inverse trig functions What is the range of possible values for inverse cosine (cos -1 )? What is the range of possible values for inverse sine (sin -1 )? What is the range of possible values for inverse tangent (tan -1 )? Evaluate the following inverse trig functions. Be sure your answer is in the correct range. cos -1 (0) sin cos sin 1 1 2

8 8 15. Solve sinusoidal equations (basic) What does it mean if an angle has a negative measure? Describe where the angle of ππ is on the Unit 3 Circle and what its cosine and sine values are. List all possible solutions to the trig equations below. Show your step in calculating the principal value. Your answers can be in degree or radian mode. cos(x) = 0.3 sin(x) = 0.6 sin(x) = -0.4 AP Only Topic: Limits Before we get to specific skills, answer the following. How is a limit different than a normal function value? What does the symbol mean mathematically? In your own words, describe what the expression lim xx 4 ff(xx) = 3 means. 16. Limits from tables (**AP only**) Complete the exercise below, and describe your reasoning.

9 9 17. Approximating limits from graphs (**AP only**) Complete the exercise below, and describe your reasoning. 18. One-sided limits from graphs (**AP only**) What does the notation xx 3 + and xx 3 mean? How is it different than xx 3? Complete the exercise below, and describe your reasoning.

10 Limits by direct substitution (**AP only**) How do you calculate a limit using algebra? Calculate the following limits. Show your steps. lim(3xx 5) lim xx 4 xx 2 xx 2 +2xx 5 xx 2 3xx 2 3 cos xx lim xx ππ xx Limits by factoring (**AP only**) What type of limit problems require factoring? How does factoring help you solve these problems? Calculate the following limits. Show your steps. 4xx+12 lim xx 3 2xx+6 lim xx 5 xx 2 7xx+10 xx 2 4xx 5 xx lim 2 2xx xx 2 xx 2 4 In order to be prepared for class on Thursday, August 10, you must complete the following: Practice all 20 skills on Khan Academy Complete the written study guide where you show your work solving example problems Complete Mastery Challenges on Khan Academy throughout the summer to keep your skills sharp