Math 124 Final Examination Winter 2023

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Student ID # Quiz Section

Professor’s Name TA’s Name

READ THE INSTRUCTIONS!

• These exams will be scanned. Write your name, student number and quiz section clearly.

• Turn off and stow away all cell phones, smart watches, mp3 players, and other similar devices.
No earbuds/headphones allowed during the exam.

• This exam is closed book. You may use one 8.5′′ × 11′′ sheet of handwritten notes (both sides
OK). Do not share notes. No photocopied or printed materials are allowed.
π √
• Give your answers in exact form unless instructed otherwise. For example, or 5 3 are exact
3
numbers while 1.047 and 8.66 are decimal approximations for the same numbers.

• You can only use a Texas Instruments TI-30X IIS calculator.

• In order to receive credit, you must show all of your work. If you do not indicate the way
in which you solved a problem, you may get little or no credit for it, even if your answer is
correct.

• This exam has 11 pages plus this cover page with 8 questions. Please make sure that your
exam is complete.

Problem Score Problem Score Problem Score

1 (12 pts) 4 (12 pts) 7 (12 pts)
2 (12 pts) 5 (12 pts) 8 (16 pts)
3 ( 12 pts) 6 (12 pts) Total
Math 124, Winter 2023 Final Examination Page 1 of 11

1. (12 total points) Compute each of the following limits. If there is no finite limit, write ∞, −∞,
or DNE (does not exist), whichever applies.
x sin x
(a) (4 points) lim
x→0 1 − cos x

x2 − 3x + 2
(b) (4 points) lim
x→1 x−1

 
ln(ln x)
(c) (4 points) lim cos
x→∞ ln x
Math 124, Winter 2023 Final Examination Page 2 of 11

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 .

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

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

Math 124, Winter 2023 Final Examination Page 3 of 11

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.

y = f ′ (x)

(a) lim f ′ (x) =

x→−3+

(b) f ′′ (−7) =

(c) List all values of x where the graph of y = f (x) has a local maximum.

(d) List all intervals where the graph of y = f (x) is concave up.

(e) List all x values where the graph of y = f (x) has a point of inflection.

(f) If f (−2) = 0, what is f (−0.5)?

Math 124, Winter 2023 Final Examination Page 4 of 11

4. (12 points) Consider the curve defined by the

implicit function x3 − 4xy + y 2 = 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.
Math 124, Winter 2023 Final Examination Page 5 of 11

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

has parametric equations

x(t) = t cos(πt) y(t) = t sin(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,

lim s(t) =
t→∞
Math 124, Winter 2023 Final Examination Page 6 of 11

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?
Math 124, Winter 2023 Final Examination Page 7 of 11

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.
4
Recall that the volume of a sphere of radius r is given by V = πr3 and its surface area is given
3
by A = 4πr2
Math 124, Winter 2023 Final Examination Page 8 of 11

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

6 6
y = f (x) = 2 − + 2
x x
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.

Continued on the next page.

Math 124, Winter 2023 Final Examination Page 9 of 11

6 6
8. (continued) Recall the function y = f (x) = 2 − + 2
x x
(c) Calculate the following limits.
(i) lim f (x)
x→∞

(ii) lim f (x)

x→−∞

(iii) lim+ f (x)

x→0

(iv) lim− f (x)

x→0

(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.
Math 124, Winter 2023 Final Examination Page 10 of 11

This page is blank. If you continued a question here, make a note on the question page so we check
it.
Math 124, Winter 2023 Final Examination Page 11 of 11

This page is blank. If you continued a question here, make a note on the question page so we check
it.