c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Exam 2 Review
1. Use differentials to approximate the value of
√
4
16.2.
2. Find the linear approximation to f (x) = (x − 2)3 at x = 4 and use it to approximate the value
of 2.053 .
3. Find the quadratic approximation to f (x) = cos x at x = π3 .
4. Calculate the following limits.
x−8
(a) lim
x→6+
1
2
x−6
e5x − e−2x
x→∞ e6x − e−4x
(b) lim
5. Find an equation of the tangent line to the graph of f (x) = xex
6. Consider the function f (x) =
5 +x
at x = 1.
x3 − 4
.
x3 − 9
(a) Show that f (x) is one-to-one.
(b) Find the inverse of f (x).
7. Given f (x) = e3x+3 − 4x3 − 2, find g 0 (3) where g is the inverse of f .
8. A 5-meter drawbridge is raised so that the angle of elevation changes at a rate of 0.1 rad/s. At what
rate is the height of the drawbridge changing when it is 2 m off the ground?
9. A dog sees a squirrel at the base of a tree 10 ft away. The dog takes off running toward the tree with
a speed of 3 ft/s. The squirrel takes off up the tree at the same time with a speed of 5 ft/s. At what
rate is the distance between them changing 2 seconds later?
10. A 25-ft long trough has ends which are isosceles triangles with height 5 ft and a length of 4 ft across
the top. Water is poured in at a rate of 15 ft3 /min, but water is also leaking out of the trough at a
rate of 3 ft3 /min. At what rate is the water level changing when the width of the water across the
trough is 2 ft?
11. Calculate the following limits.
cos x + tan 9x − 1
x→0
sin x
x cos 7x sin 4x
(b) lim
x→0
3 sin2 10x
(a) lim
12. Find the values of x, 0 ≤ x ≤ 2π where the tangent line to f (x) = sin2 x + cos x is horizontal.
3
13. Find y 0 for the equation sin(3x − y) + xy 2 = ex y .
14. Find the slope of the tangent line to the graph of 5x2 y − y 3 = −12 at the point (1, 3).
15. For what value(s) of a are the curves
x2 y 2
5
+
= and y 2 − 4x2 = 5 orthogonal at the point (1, 3)?
2
a
9
4
16. Find a tangent vector to the curve r(t) =< cos3 3t, 2 sin 5t + cos 4t > at the point where t = π4 .
1
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
17. The position of an object is given by the function r(t) =<
√
t2 + 8t,
t
>.
3t − 2
(a) What is the speed of the particle at time t = 1?
(b) What is the acceleration at time t = 1?
18. Find f (29) (x) where f (x) = e−4x + cos 3x.
19. Consider the curve x = t2 + 6t, y = 2t3 − 9t2 .
(a) Find the slope of the tangent line at the point (−5, −11).
(b) Find the points on the curve where the tangent line is horizontal or vertical.
20. Find an equation of the tangent line to the curve x = tan 2t +
4
, y = (t2 + 3t + 2)4 − (t + 1)5/2
(t + 1)2
at the point where t = 0.
21. Suppose F (x) =
x
f
f0
g
g0
0
2
5
2
−4
1
1
6
1
24
g(7x + 1) + f (5x + g(cos x)) + g(f (3x)). Use the table of values below to find F 0 (0).
p
3
2
7
3
−3
−1
22. Show that the function f (x) = e2x cos x satisfies the differential equation y 00 − 4y 0 + 5y = 0.
23. Differentiate the following.
√
(a) f (x) = tan2 5x + sec( 3x2 − x3 )
!8
3t2 + 5t
(b) g(t) =
4t3 − t2
√
x
(c) h(x) = ( x62 − 4 8x6 − x)5 (cot 9x + ee )
2