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Math 124

Final Examination

Winter 2023

1. (12 total points) Compute each of the following limits. If there is no finite limit, write ∞, −∞, or DNE (does not exist), whichever applies.

(a) (4 points)

(b) (4 points)

(c) (4 points)

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

(a) (4 points) y = (1 + √ t)20(1 + t)23.

(b) (4 points) y = (sin x + tan3 x)1/3.

(c) (4 points) y = (1 + cos x + e x )sin x.

3. (12 points) The function f(x) is differentiable everywhere except at x = −3. Answer the questions based on the graph of y = f′(x), the derivative of f(x), shown below.

4. (12 points) Consider the curve defined by the implicit function x3 − 4xy + y2 = 0 whose graph is shown on the right.

(a) Find all points (a, b) on the curve where the tangent line is vertical.

(b) Check that the point (3, 3) lies on the curve. Use a linear approximation to estimate the x-coordinate of a point on the curve with y-coordinate equal to 2.95.

5. (12 points) A particle is moving in the plane and has parametric equations

x(t) = t cos(πt)                    y(t) = tsin(πt)

where t ≥ 0. The path of the particle during the time interval 0 ≤ t ≤ 2 is plotted on the right.

(a) What is the horizontal velocity of the particle at time t?

(b) What is the vertical velocity of the particle at time t?

(c) What is the equation of the tangent line to the path when the particle crosses the negative y-axis?

(d) If s(t) is the speed of the particle at time t,

6. (12 points) The top of a ladder slides down a vertical wall at a rate of 0.25 meters per second. At the moment when the bottom of the ladder is 5 meters from the wall, it slides away from the wall at a rate of 0.6 meters per second. How long is the ladder?

7. (12 points) A lump of clay of volume 1000 cubic centimeters is used to make a cube and a sphere. Find the dimensions of the cube and the sphere that would give the maximum and the minimum total surface areas.

Recall that the volume of a sphere of radius r is given by V = 4/3πr3 and its surface area is given by A = 4πr2

8. (16 points) Let f(x) be the function

on the domain of all non-zero real numbers.

(a) Find all intervals over which f(x) is decreasing.

(b) Find all intervals over which f(x) is concave down.

8. (continued) Recall the function

(c) Calculate the following limits.

(d) Sketch the graph f(x) using the grid below. Clearly label the (x, y) coordinates of all critical points and all points of inflection.