Download Final Exam - UCR Math Dept.

University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9A - First Year of Calculus
Sample 1
Instructions: This exam has a total of 200 points. You have 3 hours.
You must show all your work to receive full credit. You may use any result
done in class. The points attached to each problem are indicated beside the
problem. You are not√allowed to use books, notes, or calculators. Answers
should be written as 2 as opposed to 1.4142135....
1. In each part, compute the limit. If the limit is infinite, be sure to
specify positive or negative infinity.
(a) (6 points)
x3 − 9x
x→−3 6 + 2x
lim
Answer: 9.
(b) (7 points)
lim+
x→0
sin(2x)
x2
Answer: +∞. Divide the top and the bottom by 2x.
(c) (7 points)
3x
lim √
2
x→−∞
4x + x + 5
Answer: −3/2. Divide the top and the bottom by x.
2. Consider the following piecewise defined function:
(
x+5
if x < 3
√
f (x) =
4 x + 1 if x ≥ 3
1
(a) (6 points) Show that f (x) is continuous at x = 3.
Answer: Show limx→3− f (x) = 8 = limx→3+ f (x).
(b) (14 points) Using the limit definition of the derivative, and computing the limits from both sides, show that f (x) is differentiable
at x = 3.
(3)
(3)
Answer: Show limx→3+ f (x)−f
= 1 = limx→3− f (x)−f
.
x−3
x−3
3. (20 points) If
y = x2 + cos(π(x2 + 1))
dy
compute dx
and find the equation for the tangent line at x0 = 1. You
may leave your answers in point-slope form.
Answer: y 0 = 2x − 2πx sin(π(x2 + 1)) and y 0 (1) = 2.
4. A projectile is launched from the ground; its height at time t is given
by the function h(t). A 16 ft. street lamp is located 5 feet away from
the launch site. At time t, the shadow of the projectile is located s(t)
feet away from the base of the lamp.
(a) (6 points) Draw a diagram illustrating the situation. Be sure to
label h(t), s(t), the height of the street lamp and the distance
from the projectile to the street lamp.
(b) (7 points) Express s(t) as a function of h(t).
Answer: Since s/(s + 5) = h/16 we get s = 5h/(16 − h).
(c) (7 points) Assuming the initial velocity of the projectile is 20 ft./s.
(in which case h(t) = −16t2 + 20t), compute the rate at which the
shadow of the projectile is moving at t = 1 s. At that time, is the
shadow moving towards the street lamp, or away from it?
Answer: From the previous part, we have
ds/dt =
80
dh
80
ds dh
=
=
(20 − 32t).
2
dh dt
(16 − h) dt
(16 − h)2
At t = 1, h = 4, so ds/dt = −20/3 ft/sec.
5. Consider the following function:
f (x) = 3x − 2 sin x + 7
2
(a) (10 points) Use the Intermediate Value Theorem to show that f (x)
has at least one zero.
Answer: Note that f (−5) < 0 and f (0) > 0.
(b) (10 points) Use the Mean Value Theorem to show that f (x) has
at most one zero.
Answer: If a < b are two different two zeroes, then f 0 (x) has a
zero in [a, b]. But f 0 (x) = 3 − 2 cos x > 0 always.
6. A curve is defined implicitly by the equation
x3 + y 3 = 6xy
(a) (10 points) Using implicit differentiation, compute
Answer: y 0 = (6y − 3x2 )/(3y 2 − 6x).
dy
.
dx
(b) (10 points) Determine the location of each point on the curve
whose tangent line has a slope of −1.
Answer: (0, 0), (3, 3).
7. You are designing an athletic field for a local high school. The athletic field will be in the shape of a rectangle of length y capped with
semicircular regions of radius r at either end. Furthermore, the entire
oval-shaped field is to be bounded by a racetrack of length 500 m.
(a) (6 points) Sketch the athletic field, labeling all relevant quantities.
(b) (7 points) Express the area of the rectangular portion of the playing field as a function of r, the radius of the semicircular caps.
Answer: A(r) = (500 − 2πr)r = 500r − 2πr2 .
(c) (7 points) Determine the dimensions of such a playing field for
which the rectangular portion is largest.
Answer: Let dA/dr = 0 we get r = 125/π. The dimensions are
r = 125/π and 250 − πr = 125.
8. Compute an antiderivative for each of the following functions:
3
(a) (10 points) f (x) = 2xx2+1
Answer: f (x) = 2x + 1/x2 . One antiderivative is x2 − 1/x.
3
(b) (10 points) h(x) = csc(2x) cot(2x)
Answer: − csc 2x/2.
9. Consider the following function:
f (x) =
2x2
(x2 − 4)
(a) (5 points) Find all vertical and horizontal asymptotes for this function. At the vertical asymptotes, specify whether the graph is
approaching positive or negative infinity.
Answer: Horizontal: y = 2. Vertical: x = 2 or x = −2. We
have:
lim f (x) = lim f (x) = 2;
x→∞
x→−∞
lim f (x) = lim − f (x) = ∞;
x→2+
x→−2
lim f (x) = lim + f (x) = −∞.
x→2−
x→−2
(b) (5 points) Determine the intervals on which the function is increasing and decreasing.
Answer: Note that f 0 (x) = −16x/(x2 − 4)2 > 0 if and only if
x > 0.
(c) (5 points) Determine the intervals on which the function is concave
up and concave down.
Answer: Note that f 0 (x) = (48x2 + 64)/(x2 − 4)3 > 0 if and only
if x2 − 4 > 0. if and only if x > 2 or x < −2.
(d) (5 points) Using your answers from parts (a), (b) and (c), sketch
the graph y = f (x).
10. Consider the following continuous function:
f (x) = x1/3 (x − 8)
defined on the closed, bounded interval [−8, 8].
4
(a) (10 points) Find all of the critical points for f (x).
Answer: Note that f (x) = x3/4 − 8x1/3 , so
f 0 (x) = 4x1/3 /3 − 8x−2/3 /3 = 4x−2/3 (x − 2)/3.
Critical points are x = 0 and x = 2.
(b) (10 points) Determine the absolute maximum and absolute minimum values for f (x) on the interval [−8, 8].
√
abAnswer: f (0) = 0, f (2) = −6 3 2, f (8) = 0, f (−8) = 32. So √
solute maximal value is 32, and absolute minimum value is −6 3 2.
5