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Math 124

Final Examination

Spring 2022

1. (15 total points) Answer the following.

(a) (6 points) Evaluate

using two different methods, neither of which is guessing the answer from a table of values.

Using first method:

Using second method:

(b) (4 points) Evaluate

(c) (5 points) Find the values of a and c so that f(x) is continuous x = 0.

2. (18 total points) Find the derivatives of the following functions. You do not have to simplify.

(a) (4 points) f(x) = 3tan(4x)

(b) (4 points) h(x) = sin(x2) cos2(3x)

(c) (5 points) y(x) = xex

(d) (5 points)

3. (12 points) The function f(x) whose graph is given below has domain all numbers except for x = 5. Answer the questions based on the graph below.

(d) Give your best estimate for f(1.85).

(e) Let g(x) = (f(x))2. Find all values of x where g′(x) = 0.

(f) Let h(x) = f(f(x)). Compute h′(3).

4. (12 points) A curve is given implicitly by the equation

x3 + y3 = 2xy.

A graph is shown on the right to help you visualize.

(a) Find the equation of the tangent line to the curve at the point (1, 1).

(b) Use linear approximation to find the value of x when y = 0.93.

(c) Find the coordinates of the point P on the curve where the tangent line is horizontal.

5. (12 points) An object is moving in the xy-plane according to the parametric equations:

x(t) = 5 cos(πt)

y(t) = 4 sin(πt)

When 0 < t < 1/2 , the location P(t) = (x(t), y(t)) of the object will be in the first quadrant, as pictured below. Let ℓ be the normal line to the trajectory at P(t) and xN the x-intercept of ℓ. (The normal line ℓ is perpendicular to the tangent line through P(t).)

(a) Write the equation of the normal line ℓ through P(t), assuming 0 < t < 1/2.

(b) Find an expression for xN as a function of t.

(c) Compute

6. (10 points) The dimensions of a rectangular box with square bottom are changing. The length of the side of the base is increasing at a constant rate of 3 ft/min and the height is decreasing at a constant rate of 5 ft/min.

(a) What is the rate of change of the volume of the box when the length of the side of the base is 4 ft and the height is 3 ft?

(b) Is the rate of change of the volume of the box increasing or decreasing when the length of the side of the base is 4 ft and the height is 3 ft? Explain.

7. (11 points) A rectangle is inscribed in the ellipse

with sides parallel to the axes, as pictured. Find the dimensions of the rectangle of maximum perimeter which can be so inscribed.

8. (15 total points) Consider the function

(a) What are the x- and y-intercepts of the graph of f?

(b) Determine the horizontal asymptotes of f.

(c) Find all critical numbers of f(x) and determine the intervals in which f is increasing and in which it is decreasing.

(d) Determine the local minima and local maxima of f.

Recall that the function is

(e) Using f′′(x), find the intervals where f is concave up and where it is concave down.

(f) List the inflection points on the graph of f.

(g) Using all of the above information, sketch the graph of f in the provided coordinate system. Mark any important points that came up in your computations.