c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
Math 151 Exam 2 Review
1. Use a linear approximation to estimate the value of
√
5
31.8.
2. Find the linear and quadratic approximations to f (x) =
1
at x = 2.
x2
3. The diameter of a sphere was measured to be 10 inches with a possible error in measurement of 0.2
inches. Use differentials to estimate the maximum error in the volume of the sphere.
1
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
4. Calculate the following limits.
x−8
(a) lim
x→6+
(b)
1
2
x−6
4e−5x + 3e6x
x→−∞ 6e6x + 3e−5x
lim
2 + e−4x
x→∞ 2 + e7x
(c) lim
5. Find an equation of the tangent line to the graph of f (x) = xex
6. Find y 00 if y = e2x cos x.
2
5 +x
at x = 1.
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
7. Consider the one-to-one function f (x) =
4x − 4
. Find the inverse of f (x).
3x + 10
8. Given f (x) = ex+1 − 4x3 , find g 0 (5) where g is the inverse of f .
9. 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?
3
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
10. Ship A is 5 km west of ship B. Ship A begins sailing south at 2 km/h while ship B begins sailing north
at 3 km/h. How fast is the distance between the ships changing 2 hours later?
11. 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 height of the water changing when the width of the water across
the trough is 2 ft?
4
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
12. Calculate lim
x→0
x cos 5x tan 4x
3 sin2 10x
13. Find the values of x, 0 ≤ x ≤ 2π, where the tangent line to f (x) = sin2 x + cos x is horizontal.
3
14. Find y 0 for the equation sin 2y + xy 2 = ex y .
15. Find the slope of the tangent line to the graph of 5x2 y − y 3 = −12 at the point (1, 3).
5
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
√
16. Consider the vector function r(t) =< 3 t2 + 8t,
t
>.
3t − 2
(a) What is the domain of the vector function?
(b) Find a unit tangent vector to the curve at the point where t = 1.
17. The position of an object is given by the vector function r(t) =< t + cos2 t, sin 5t + cos 4t >.
(a) What is the speed of the particle at time t = π?
(b) What is the acceleration at time t = π?
6
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
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.
7
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
20. Find an equation of the tangent line to the curve x =
4
, y = (t2 + 3t + 2)4 at the point where
(t + 1)2
t = 0.
21. Suppose F (x) =
F 0 (x).
p
3
g(f (7x)) + f (5x + g(csc x)) where f (x) and g(x) are differentiable functions. Find
22. For functions f and g, we are given that f (1) = 2, f 0 (1) = 5, g(1) = 3, and g 0 (1) = 4. Suppose that
U (x) = f (x2 )eg(x) . Find an equation of the tangent line to U (x) at the point where x = 1.
8
c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
23. Given that y = 3x + 7 is the tangent line to the graph of f (x) = ax2 + bx at the point were x = 2,
find the values of a and b.
24. Find the values of a and b so that the following function is differentiable everywhere.
(
f (x) =
x2 + ax + b
ex + cos x
x<0
x≥0
25. Differentiate the following.
(a) f (x) = tan3
√
+ sec( 3 3x2 − x3 )
3t2 + 5t
4t3 − t2
(b) g(t) =
(c) h(x) =
x
6
1
4x
−
x2
5
!5/2
x
(cot 9x + ee )
9