Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Math 124

Final Examination

Autumn 2022

1. (14 total points) Determine each of the following limits. If there is no finite limit, write ∞, −∞, or DNE (does not exist), whichever applies.

(a) (5 points)

(b) (4 points)

(c) (5 points)

2. (15 total points) Find the derivatives of the following functions. You do not have to simplify.

(a) (5 points)

(b) (5 points)

(c) (5 points)

3. (12 points) Answer the questions based on the graph of f(x) given on the right. The domain of f is the set of all numbers.

Let g(x) = f (f(x)) and let h(x) = (f(x))2.

(a) Compute f′(2.5).

(b) Compute g′(5).

(c) Find all critical numbers for g(x). Is g differentiable everywhere?

(d) Compute

(e) Find the intervals where h is increasing.

4. (12 points) Consider the curve defined by the equation y = x2y.

(a) Confirm that (1, 1) is on the curve.

(b) Through linearization approximate the y-coordinate on the curve when x = 1.05.

(c) Is this an overestimate or an underestimate? Justify your answer.

5. (11 points) Consider the parametric curve defined by the equations

(a) Find the slopes of the tangent lines to the curve at t = −1 and at t = 1.

(b) The curve crosses itself at the point where t = −1 and t = 1. What is this point?

(c) Suppose we zoom in on the curve at the point from part (b). Which of the following pictures most resembles what the curve looks like near this point? Explain using derivatives why you chose your answer.

(d) For the picture you chose in part (c), draw arrows on the picture to indicate the direction the curve is traced as t increases. Use the values of dx/dt and dy/dt when t = −1 and t = 1 to explain your answer.

6. (10 points) Retta runs at a speed of 4 meters per second around a circular track of radius 50 meters.

At the center of the track is a lightbulb, and 10 meters away from the track is a wall.

When Retta is 35 meters from the wall, how fast is her shadow moving along the wall?

7. (11 points) Copper is standing at the westernmost point of the lake with radius 120 meters when he spots a duck across. To get to the duck, he can swim or run around the lake. He swims at a speed of 6 meters per second and runs at a rate of 12 meters per second. If he swims to point B in a straight line and runs around the lake from B to C, find the angle θ shown which will minimize his time to reach the duck. Make sure you justify using calculus why the answer you found gives the minimum time. Note: The picture is not to scale.

8. (15 points) Consider the function

(a) What is the domain of the function f(x)?

(b) Find the horizontal and vertical asymptotes of f(x), if any.

(c) Find the critical numbers of f(x).

(d) Find the intervals where f(x) is increasing and the intervals where f(x) is decreasing.

(e) Find the local maximum and the local minimum values of f(x) (round your answer to 2 decimal places).

Recall that the function is

(f) Find the intervals of concavity and the inflection points of f(x) (for inflection points, round your answer to 2 decimal places).

(g) Sketch the graph of y = f(x) on the axis provided below. Be sure to include asymptotes in your picture as well as to mark the coordinates of critical and inflection points.