Calculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School. Calculus Summer Math Study Guide

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1 1 Calculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School Name: Calculus Summer Math Study Guide After you have practiced the skills on Khan Academy (list available on bdcs.org/summer18), complete the following study guide. Be sure to show all work and describe your reasoning, as this study guide should be a resource for you at the beginning of the school year. If you have any questions, be sure to contact me at kbraun@bdcs.org. I will reply within 48 hours Monday-Friday. 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 non-ap 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. a. f(x) = 2x+8 5 b. g(x) = 3x Transforming Functions Match the functions below to the following transformations: shift right; shift left; shift up; shift down; vertical stretch; vertical compression; vertical reflection (reflect over x-axis); horizontal reflection (reflect over y-axis) a. g(x) = f(x + 2) b. h(x) = f(x) 4 c. k(x) = f(x 3) d. q(x) = f(x) + 7 e. r(x) = 3f(x) f. p(x) = 1 f(x) g. v(x) = f(x) h. 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. a. f(g(4)) b. g(f(4)) c. f(g(x)) d. g(f(x)) 4. Evaluate Composite Functions: Graphs & Tables Explain your reasoning as you solve these two problems from Khan Academy. a. b. 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. a. log 4 64 b. log c. log 2 1, Evaluate Logarithms (Advanced) i. What exponent is equivalent to the square root function? ii. What does a negative exponent do to an expression? For example, what is 7-2? iii. What is true of any number raised to the power of zero? Evaluate the logarithms below. Justify that each answer is correct. 1 a. log 16 4 b. log 5 c. log 8 32 d. log

4 4 9. Relationship Between Exponentials & Logarithms What is true of the relationship between an exponential function and a logarithmic function? Given what you said in Skill #1, what does this mean about these functions? Complete the following exercises. Describe your reasoning below. a. b. c. Re-write the expression 3 4 = 81 in logarithmic form. d. Re-write the expression log 2 16 = 4 in exponential 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 logarithmic expression. a. log(4) + 2log(6) b. log(10) log(2) c. 3log(6) log(9)

5 5 11. Solve Exponential Equations using Logarithms: Base-10 & Base-e i. When you see log with no base written, what is the base understood to be? ii. 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. a t = 20 b. 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 y = a sin(b(x c)) + d, describe what each term means on the graph and how you calculate the following: i. Amplitude: ii. Period: iii. Midline: Graph the following trig functions. Be sure to label the amplitude, period, and midline. a. y = 3sin(4x) + 1 b. y = 2cos (πx) Evaluate Inverse Trig Functions i. What is the range of possible values for inverse cosine (cos -1 )? ii. What is the range of possible values for inverse sine (sin -1 )? iii. 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. a. cos 1 (0) b. sin 1 ( 3 2 ) c. cos 1 ( 2 2 ) d. 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 π 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. a. cos(x) = 0.3 b. sin(x) = 0.6 c. 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 statement lim x 4 f(x) = 3 means. 16. Limits from Tables (**AP only**) Complete the exercise below, and describe your reasoning. a.

9 9 17. Approximating Limits from Graphs (**AP only**) Complete the exercise below, and describe your reasoning. a. 18. One-Sided Limits from Graphs (**AP only**) What does the notation x 3 + and x 3 mean? How are they different than x 3? Complete the exercise below, and describe your reasoning. a.

10 Limits by Direct Substitution (**AP only**) How do you calculate a limit using algebra? Calculate the following limits. Show your steps. a. lim x 4 (3x 5) b. lim x 2 +2x 5 x 2 x 2 3x 2 3 cos x c. lim x π Limits by Factoring (**AP only**) What types of limit problems require factoring? How does factoring help you solve these problems? Calculate the following limits. Show your steps. a. lim x 3 4x+12 2x+6 b. lim x2 2x 15 x 5 x 2 4x 5 c. lim x2 2x x 2 x 2 4

11 11 Answer Key 1. To solve for an inverse function algebraically, switch x and y, then isolate y. a. f 1 (x) = 5x 8 2 b. g 1 (x) = x a. Horizontal shift left two units b. Vertical shift down four units c. Horizontal shift right three units d. Vertical shift up seven units e. Vertical stretch by a factor of three f. Vertical compression by a factor of one-half g. Vertical reflection/reflection over the x-axis h. Horizontal reflection/reflection over the y-axis 3. The expression f(g(x)) means using g(x) as the input of f(x) a. f(g(4)) = 59 b. g(f(4)) = 125 c. f(g(x)) = 3x d. g(f(x)) = 9x 2 6x a. To evaluate g(h( 3)), locate 3 in the input row for h(x) and determine the corresponding output as 2. Then, locate 2 on the input row for g(x) and determine the corresponding output of 7. Therefore, g(h( 3)) = 7. b. To evaluate (g f)( 2), locate -2 on the x-axis and determine the corresponding output of f(x) as 0. Then, locate 0 on the x-axis and determine the corresponding output of g(x) as 8. Therefore, (g f)( 2) = Positive intervals are x-values for which the function has an output greater than zero. Negative intervals are x-values for which the function has an output less than zero. a. Positive intervals: {( 3.5, 1), (1, 3.5)} b. Negative intervals: {( 5, 3.5), ( 1,1), (3.5,5)} 6. Increasing intervals are x-values for which the function has a positive rate of change. Decreasing intervals are x-values for which the function has a negative rate of change. a. Increasing intervals: {( 4.5, 2.25), (0,2.25), (4.5,5)} b. Decreasing intervals: {( 5, 4.5), ( 2.25,0), (2.25,4.5)} 7. To evaluate logarithms, determine what exponent would cause the base to yield an output matching the argument. In the given example, an exponent of 3 would cause the base of 2 to yield an output of 8, which is the argument in question. Therefore, log 2 8 = 3. a. log 4 64 = 3 because 64 = 4 3 b. log = 4 because 625 = 5 4 c. log 2 1,024 = 10 because 1,024 =

12 12 i. An exponent of one-half is equivalent to the square root operation ii. A negative exponent causes division by the base instead of multiplication by the base. Therefore 7 2 = 1 or iii. Any exponent raised to the power of zero is one. b. log 16 4 = 1 because 16 = c. log 5 = 3 because = d. log 8 32 = 5 because = 32 or 8 5 = 32 3 e. log 8 1 = 0 because 8 0 = 1 9. Exponential functions and logarithmic functions are inverses. This means that if you switch the x and y-values in an exponential function and isolate y, you will have the logarithmic function. a ; 9 b. {(1,0), (2,1), (4,2), (8,3), (16,4)} c. 4 = log 3 81 d. 16 = The properties of logarithms i. log b 1 = 0 ii. log b b = 1 iii. log b (mn) = log b m + log b n 11. iv. log b ( m n ) = log b m log b n v. log b (m p ) = p log p m a. log 144 b. log 5 c. log 24 i. Base ten ii. Base e (Euler s number) b. t = 1 20 log c. x = ln

13 The cosine function is the x-value and the sine function is the y-value. 13. i. Amplitude: the vertical distance from the midline to a maximum or minimum value ii. Period: the horizontal distance required for the function to complete one full revolution iii. Midline: the imaginary horizontal line that passes through the middle of the function s maximum and minimum points a.

14 14 b. 14. i. Range of inverse cosine: [0, π] ii. Range of inverse sine: [ π 2, π 2 ] iii. Range of inverse tangent: [ π 2, π 2 ] a. cos 1 (0) = ± π 2 b. sin 1 ( 3 ) = π 2 3 c. cos 1 ( 2 ) = 3π 2 4 d. sin 1 ( 1 ) = π A negative angle in standard position has a terminal side rotated clockwise from the x-axis. Therefore, an angle of π is coterminal with a positive angle measure of 5π and has a cosine 3 3 value of 1 3 and a sine value of. 2 2 a. x or rad b. x or rad c. x or rad 16. A limit in math is a term that describes where a function is going at an x-value, regardless of where the function actually is at that value. The symbol indicates where the x-value is approaching. The statement lim f(x) = 3 is read the limit of f(x) as x approaches four is x 4 three and means that as the function gets closer to an x-value of four from either side, the y- value gets closer and closer to three. a. lim f(x) = 4.18; Even though the function equals 0.2 at x = 6, the function x 6 approaches 4.18 as it gets closer to 6 from either side. 17. a. lim g(x) = 1.5; Even though the function equals 2.4 at x = 2, the function x 2 approaches 1.5 as it gets closer to 2 from either side. 18. The notation x 3 + means as x approaches three from the right. The notation x 3 means as x approaches three from the left. These are different than x 3, which can only be answered if the left- and right-hand limits agree. a. lim h(x) = 6; Even though the function equals 2 at x = 3, from the left-hand side x 3 the function approaches a y-value of 6 as x gets closer to 3.

15 Limits by direct substitution can be calculated by evaluating the function (plugging in) for that value. a. lim(3x 5) = 12 x 4 b. lim x 2 x 2 +2x 5 3 cos x c. lim x π 1 = 5 x 2 3x 2 8 = Limits of rational functions often require factoring, particularly when the limit is of a hole in that function. This method allows you to simplify the function and calculate the y-coordinate of the hole, which answers the question. a. lim x 3 4x+12 2x+6 = 2 b. lim x2 2x 15 = 4 x 5 x 2 4x 5 3 c. lim x2 2x = 1 x 2 x 2 4 2