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Math 151 WIR, Spring 2014, Benjamin
Aurispa
Math 151 Exam 2 Review
1. Use a linear approximation to estimate the value of
√
5
31.8.
2. Find the linear approximation to f (x) = (x − 3)3 at x = 5 and use it to approximate the value
of 2.053 .
3. Find the quadratic approximation to the function f (x) = tan x at x = π3 .
4. (a) Find the differential dy for y =
2
.
x2
(b) Use differentials to approximate the value of
2
.
2.12
5. 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.
6. Suppose the linear approximation to the function f (x) at the point where x = 1 is given by L(x) =
3x + 1. Find the linear approximation to the function g(x) = (f (x) + x)3 at x = 1.
7. Calculate the following limits.
x−8
x−8
1 x−6
x→6+ 2
5 + e−4x
(c) lim
x→∞ 2 + e7x
(b) lim
(a) lim
x→6−
1
2
x−6
8
x→−∞ 2 + e9x
(d) lim
4e−5x + 3e6x
x→−∞ 6e6x + 3e−5x
(e) lim
8. Find an equation of the tangent line to the graph of f (x) = xex
5 +x
at x = 1.
9. For what values of r does the function y = erx satisfy the equation 2y 00 − 3y 0 + y = 0?
10. Find y 00 if y = e2x cos x.
11. 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?
12. 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?
13. 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 rising when the height of the water is 2 ft?
x cos 5x tan 4x
x→0
3 sin2 10x
14. Calculate lim
15. Find the values of x, 0 ≤ x ≤ 2π, where the tangent line to f (x) = sin2 x + cos x is horizontal.
3
16. Find y 0 for the equation sin 2y + xy 2 = ex y .
17. Find the slope of the tangent line to the graph of
x2
2
+ 5x2 y −
y3
27
= 61 at the point (2, 3).
√
t+2
18. Consider the vector function r(t) =< 3 t2 + 8t,
>. Find a unit tangent vector to the curve at
3t − 2
the point (9, 3).
1
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
19. The position of an object is given by the vector function r(t) =< t cos 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?
20. Find f (29) (x) where f (x) = e−4x − sin 3x.
21. 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.
22. 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.
0
0
23. For functions
f and g, we are given that f (1) = 2, f (1) = 5, g(1) = 3, and g (1) = 4. Suppose that
U (x) = f x1 eg(x) . Find an equation of the tangent line to U (x) at the point where x = 1.
24. 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.
25. Find the points on the graph of f (x) = x2 − 3x where the tangent lines at these points also pass
through the point (1, −11).
26. 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
27. A particle has position function s(t) = t3 − 3t2 , t ≥ 0.
(a) At what rate is the particle’s velocity changing when t = 3?
(b) What is the total distance traveled by the particle during the first 4 seconds?
28. Differentiate the following functions.
(a) h(x) =
3
4x2
−
(b) f (x) = cot6
(c) g(t) =
√
5
x
5
1
√
4 x
+ 3e4
√
3
x
2
3
6 + sec( 3x − x )
3t2 + 5t
4t3 − t2
+
!5/2
t
(d) v(t) = csc(sin(9t + ee ))
(e) F (x) = g(f (7x)) + f (5x + g(tan x)) where f (x) and g(x) are differentiable functions.
2