Math 151 Week in Review
Review For Exam 2 (3.3-4.2)
Monday Oct 25, 2010
Instructor: Jenn Whitfield
18. A particle follows the path described by x =
t4 − 4t3 and y = 3t2 − 6t.
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
(a) Find the equation of the tangent line
at the point (5,9).
(b) Find all point(s) on the curve where
the tangent line is vertical or horizontal.
2x − 3−x
.
1. Compute lim x
x→∞ 4 + 5−x
(c) When is the particle moving to the
right? Left? Up? Down?
x−1
2. Find lim 5 2−x .
x→2+
2x
3. Find f ′ (1) for f (x) = ee .
4. For what values of r does the function y =
erx satisfy the equation y ′′ + 2y ′ − 3y = 0?
√
5. Find g′ (t) for g(t) = sin(e
t
+ 1).
6. Given f (x) = 2x3 + 6x + 2 and g(x) is the
inverse of f (x):
(a) Compute g(10).
(b) Compute g′ (10).
7. Find the inverse of f (x) =
√
3
8. Find the inverse of f (x) =
x−2
.
x+2
2x + 5.
9. Let g(x) = f−1 (x)
be the inverse of f (x) =
πx
2
3 + x + tan
, −1 < x < 1. Find g′ (3).
2
sin(3x)
.
x→0
5x
10. Find lim
19. A policeman with radar is sitting 0.3 miles
off a straight road. He observes a car moving
along the road. At the instant when the
distance between the car and the policeman
is 0.5 miles, that distance increases at a rate
of 50 mph. How fast is the car moving at
that instant?
20. Two sides of a triangle are 4m and 5m in
length and the angle between them is increasing at a rate of 0.06 rad/sec. Find the
rate at which the area of the triangle is increasing when the angle between the sides
of fixed length is π3
21. A cylindrical can is undergoing a transformation in which the radius and height are
varying continuously with time t. The radius is increasing at a rate of 4 inches per
minute while the height is decreasing at a
rate of 10 inches per minute. Find the rate
of change of the volume when the radius is
3 inches and the height is 5 inches.
2
11. Find f ′ (x) for f (x) = (x2 + 1)3 etan(x ) .
12. Let f (x) =
p
1 + xe−2x . Compute f ′ (0).
13. Find the parametric equations of the tangent line to the parametric curve x = cos t+
π
cos(2t), y = sin t + sin(2t) for t = .
2
14. Suppose that w = u ◦ v, u(0) = 1,
v(0) = 2, u′ (0) = 3, u′ (2) = 4, v ′ (0) = 5, and
v ′ (2) = 6. Find w′ (0).
15. Find
dy
dx
for cos(x − 2y) = x3 y 3 + 1.
16. Find the tangent line to the graph
of
π
y 2 sin(2x) = 8 − 2y at the point
,2 .
4
17. Given r(t) = h2 cos t, 3 sin ti, find the position, velocity, and acceleration vector to the
π
curve at t = .
3
22. Use the linear approximation
with a = 64
√
3
to estimate the number 70.
√
23. Find the linearization for f (x) = √ x + 3 at
x = 1 and use it to approximate 3.98
24. Find the quadratic approximation to
1
f (x) =
at x = 1.
x+2