AP Calculus

Summer Assignment

Name_____________________________

This assignment will be due the first day of class.

Late assignments will have ten points deducted for each day late.

Show ALL WORK for ALL PROBLEMS. Credit will not be given to

those answers lacking written work.

Use the internet and other resources to aid you in solving difficult

problems.

1) Consider the equation 𝑦 = √9 − 𝑥2.

a. Sketch a graph.

b. Find the domain.

c. Find the range.

d. Identify any symmetry.

e. Explain the relationship between the equation and the graph.

For numbers 2 and 3, use an algebraic method to find any intercepts. 2) 𝑥2𝑦 − 𝑥2 + 4𝑦 = 0

3) 𝑦 = 2𝑥 − √𝑥2 + 1

4) Use an algebraic method to find the points of intersection of the graphs of the following equations.

𝑦 = 𝑥4 − 𝑥3 + 2𝑥2 − 1 𝑦 = −𝑥3 + 7𝑥2 − 5

5) Find the equation of the line tangent to the circle(𝑥 − 1)2 + (𝑦 − 1)2 = 25 at the point (4, −3). Hint. Lines that are tangent to a circle are perpendicular to the radius that connects the center of the circle to a given point on the perimeter of the circle. See example below.

6) Use algebraic methods to match the equation or equations with the given characteristic.

a) Symmetric with respect to the y-axis

b) Three x-intercepts

c) Symmetric with respect to the x-axis

d) (-2, 1) is a point on the graph.

e) Symmetric with respect to the origin

f) Graph passes through the origin.

7) A line is represented by the equation 𝑎𝑥 + 𝑏𝑦 = 4

a) When is the line parallel to the x-axis?

b) When is the line parallel to the y-axis?

c) Give values for a and b such that the line is parallel to 𝑦 =2

3𝑥 + 3.

d) Give values for a and b such that the line coincides with the graph of 5𝑥 + 6𝑦 = 8.

8) Specify the sequence of transformations that will yield each graph of h from the graph of the function 𝑓(𝑥) = sin 𝑥. Then sketch the graph.

ℎ(𝑥) = − cos(𝑥 − 1)

9) Use the graphs of f and g to evaluate each expression. If the result is undefined, explain why.

a) (𝑓 ° 𝑔)(3)

b) 𝑔(𝑓(5))

c) (𝑔 ° 𝑓)(−1)

d) 𝑔(𝑓(2))

e) (𝑓 ° 𝑔)(−3)

f) 𝑓(𝑔(−1))

g) Sketch a graph of 𝑓(𝑥 + 1).

h) Sketch a graph of −2𝑔(𝑥).

i) Sketch a graph of(𝑓 ° 𝑔)(𝑥).

10) Water runs into a vase of height 30 centimeters at a constant rate. The vase is full after 5 seconds. Use the information and the shape of the vase shown to answer the questions if d is the depth of the water in centimeters and t is the time in seconds

a) Explain why d is a function of t.

b) Determine the domain and range of the function.

c) Sketch a possible graph of the function.

d) Use the graph in part c to approximate 𝑑(4). What does this represent?

11) A v8 car engine is coupled to a dynamometer, and the horsepower y is measured at different engine speeds x (in thousands of revolutions per minute). The results are shown in the table.

a) Use the regression capabilities of the graphing utility to find a cubic model for the data.

b) Use the graphing utility to plot the data and graph the model.

c) Use the model to approximate the horsepower when the engine is running at 4500 revolutions per minute.

12) Findthe exact solution and its approximation rounded to three decimal places.

a) 42𝑥+3 = 5𝑥−2 b) ln(𝑥2 + 4) − ln(𝑥 + 2) = 2 + ln (𝑥 − 2)

13) Find all exact solutions of the equation.

sin 4𝑥 cos 𝑥 = sin 𝑥 cos 4𝑥

14) You are in a boat 2 miles from the nearest point on the coast. You are to go to point Q located 3 miles down the cost and 1 mile inland. You can row at 2 miles per hour and walk at 4 miles per hour. Write the total time T of the trip as a function of x.

15) Complete the table and use the result to estimate the limit.

a)

b)

16) Use the graph of the function f to decide whether the value of the given quantity exists. If it

does, find it. If not, explain why.

a) 𝑓(1)

b) lim𝑥→−2

𝑓(𝑥)

c) 𝑓(4)

d) lim𝑥→4

𝑓(𝑥)

17) Consider the general function 𝑓(𝑥).

a) If 𝑓(2) = 4, can you conclude anything about the limit of 𝑓(𝑥) as x approaches 2? Explain your reasoning.

b) If the limit of 𝑓(𝑥)as x approaches 2 is 4, can you conclude anything about 𝑓(2)? Explain your reasoning.

18) Find The limit if it exists. a)

b)

c)

d)

e) Given that 𝑓(𝑥) = 2𝑥4 + 𝑥2 − 5, find

19)

20) Discuss the continuity of the function.

21) Every day you dissolve 28 ounces of chlorine in a swimming pool. The graph shows the amount of chlorine 𝑓(𝑡) in the pool after t days.

22) Prove that if f is continuous and has no zeros on the [𝑎, 𝑏], then either

23) Find the limit if it exists. a)

b)

24) Given a polynomial𝑝(𝑥), is it true that the graph of the function given by 𝑓(𝑥) =𝑝(𝑥)

𝑥−1has a

vertical asymptote at 𝑥 = 1? Why or why not?

25) Explain your reasoning.