Math 124 Final Examination Autumn 2016
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Professor’s Name TA’s Name
• Turn off all cell phones, pagers, radios, mp3 players, and other similar devices.
• 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 materials are allowed.
π √
• Give your answers in exact form, for example or 5 3.
3
• You can use only 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.
• Place a box around your answer to each question.
• If you need more room, use the backs of the pages and indicate that you have done so.
• Raise your hand if you have a question.
• This exam has 9 pages, plus this cover sheet. Please make sure that your exam is complete.
Question Points Score Question Points Score
1 12 5 12
2 12 6 12
3 12 7 12
4 12 8 16
Total 100
Math 124, Autumn 2016 Final Examination Page 1 of 9
1. (12 total points) Find the derivative of the following functions.
2
(a) (4 points) g(x) = e−x arctan x
(b) (4 points) Suppose that f (0) = π /4 and f ′ (0) = 3. Let h(x) = ln(tan( f (x))). Compute h′ (0).
(c) (4 points) y = (3 + 2 sin x)3x
Math 124, Autumn 2016 Final Examination Page 2 of 9
2. (12 total points) Compute the following limits. If you apply L’Hôpital’s rule then you must show that
you have checked the hypotheses.
p
(a) (4 points) lim x4 + 7x2 − x2
x→∞
1 − t + lnt
(b) (4 points) lim
t→1 1 + cos(π t)
x
(c) (4 points) lim
x→∞ 2x − sin x
Math 124, Autumn 2016 Final Examination Page 3 of 9
3. (12 total points) An object is moving along an ellipse. Its location is given by the parametric equations
x(t) = 1 + 2 cost y(t) = 2 + 4 sint
In this problem we take 0 ≤ t ≤ 2π .
(a) (3 points) Find a formula that gives the slope of the tangent line to the path at time t as a function
of t.
(b) (4 points) Find the equation of the tangent line at t = π3 .
(c) (5 points) Find all the values of t when the tangent line is perpendicular to the line x − 2y = 3.
Math 124, Autumn 2016 Final Examination Page 4 of 9
4. (12 total points) For this problem, refer
to the pictured graph of the function
y = f (x) on the interval [-2,12].
f (x) − f (7)
(a) (2 points) lim =
x→7 x−7
(b) (2 points) lim f ′ (x) =
x→2
(c) (2 points) lim f ′′ (x) =
x→2
f (x)
(d) (2 points) lim =
x→2 x
(e) (2 points) Circle the smallest number in this list:
f ′ (0) f ′ (1) f ′ (7) f ′ (9) f ′ (11)
(f) (2 points) Give an interval (a, b) on which f ′ (x) is increasing.
Math 124, Autumn 2016 Final Examination Page 5 of 9
5. (12 total points) Consider the plane curve
2x4 − 4xy + y2 = 16.
(a) (6 points) Use the tangent line approximation at (0, 4) to estimate the value of y when x = −0.04.
2
d y
(b) (6 points) Find the second derivative dx 2 at (0, 4) and use this to decide if the tangent line
approximation is an overestimate or an underestimate near (0, 4). Explain your reasoning.
Math 124, Autumn 2016 Final Examination Page 6 of 9
111
000
000
111
6. (12 points) The side wall of a building is to be braced by a beam
000
111
which must pass over a parallel wall 8 feet high and 1 foot from the 000
111
building. Find the length L of the shortest beam that can be used. 000
111
000
111
000
111
L
Verify that your answer is a minumum.
000
111
000
111
8
000
111
000
111 1
Math 124, Autumn 2016 Final Examination Page 7 of 9
7. (12 points) A pool is 15 meters wide and 22 meters long. The cross section is in the shape of a right
trapezoid, one of the parallel sides being the length of the pool, and the other one equal to 6 meters.
At its deepest point the pool is 4 meters deep. The pool and its cross section are pictured below. Water
is pumped into the pool at a rate of 7 cubic meters per minute. How fast is the water level rising when
the water level is 3 meters measured at its deepest end?
Math 124, Autumn 2016 Final Examination Page 8 of 9
8. (16 total points) Let f (x) be the function
12 12 4
y = f (x) = − 2+ 3
x x x
on the domain of all non-zero real numbers.
(a) (4 points) Find all intervals over which f (x) is decreasing.
(b) (4 points) Find all intervals over which f (x) is concave down.
Math 124, Autumn 2016 Final Examination Page 9 of 9
12 12 4
8. (continued) Recall the function y = f (x) = − 2+ 3
x x x
(c) (4 points) Calculate the following limits.
1. lim f (x)
x→∞
2. lim f (x)
x→−∞
3. lim f (x)
x→0+
4. lim f (x)
x→0−
(d) (4 points) Sketch the graph f (x) using the grid below. Clearly label the (x, y) coordinates of all
critical points and all points of inflection.
