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Math 124

Final Examination

Autumn 2021

1. (12 total points) Answer the following.

(a) (4 points) Evaluate

(b) (4 points) Evaluate

(c) (4 points) Use the limit definition of the derivative to find

2. (12 total points) Find the derivatives of the following functions. You do not need to simplify your answers.

(a) (4 points)

(b) (4 points) f(x) = ln (sin(ln(x)))

(c) (4 points)

3. (16 points) The function whose graph is shown below has domain all real numbers except x = −10 and the value f(−7) = −3. For limit questions your answer must be a number, −∞, ∞, or DNE. Make your best estimate in the case you cannot clearly read the numbers from the graph. You do not have to show your work.

4. (11 points) The point (1, 1) lies on the implicitly defined curve

(a) Use linearization to approximate the coordinates (x, y) on the curve when x = 1.01.

(b) Is the approximation for y an over-estimate or an under-estimate? Justify your answer.

5. (10 points) In the picture shown, the line is tangent to the parametric curve

at the point (3, 8). Find the coordinates of the point P, where the tangent line intersects the curve again.

6. (12 points) Label the sides of an isosceles triangle as pictured. The side w is increasing at a rate of 3 ft/min while the sides ` are decreasing at a rate of 2 ft/min. Find how fast the angles shown are changing when w = 10 feet and θ = π/4 radians.

7. (10 points) You have a circular sheet of paper, measuring 6 inches in diameter. From it, you want to make a conical drinking cup by folding together two radial lines of the circle. What is the maximum volume of a cone you can make in this way?

Recall that the volume of a cone of radius r and height h is V = 1/3πr2h.

Make sure to justify that you have maximized the volume.

8. (17 total points) Consider the function

(a) (3 points) Find the x and y intercepts of the graph of y = f(x).

(b) (3 points) Find the vertical and horizontal asymptotes, if any.

(c) (4 points) Find the critical numbers for f(x) and determine if each gives a local minimum, a local maximum, or neither.

Recall that the function is

(d) (4 points) Find the inflection point(s) of y = f(x). In which interval(s) is the graph concave down?

(e) (3 points) Sketch the graph of y = f(x) on the axes provided below. Mark the coordinates of any local maximum, local minimum or inflection points. Make sure your picture matches the information you provided in parts (a)-(d).