Math 171 — Calculus I Spring, 2016
Practice Questions for Exam II
x−2 − 14
Problem 1. What does the limit lim
represent?
x→2 x − 2
1
d
1
D) −
A)
|
x=2
2
dx x2
E) The limit does not exist.
B) 0
C) ln(2)
Problem 2.
functions.
Use the definition of the derivative to find the derivative of the following
A) g(z) = 7z 2 + 2z − 3
B) f (x) =
√
2x + 5
C) f (t) =
3
t−1
√
Problem 3. Given L(h) = h2 − s2 where s is constant, find L0 (h) using the limit
definition of the derivative. If L has units km and h has units km what are the units of
L0 (h). For which values of h is L0 undefined?
Problem 4. Consider the function f (x) = x4 − 8x2 + 20x + 10. Use the Intermediate
Value Theorem to show that the graph of f has a horizontal tangent line between x = −3
and x = −2.
Problem 5.
Consider the graph below of the function f (x) on the interval [0, 5].
1. For which x values would the derivative f 0 (x) not be defined?
2. Sketch the graph of the derivative
function f 0 .
Problem 6. The Monod Growth function is given by
G(S) = µ
S
K +S
where µ and K are constant.
A) Find G0 (S).
B) Find the equation of the line tangent to the curve y = G(S) at S = 0.
C) Find lim G0 (S).
S→∞
1
2
Problem 7.
A) Find the equation of the line tangent to the curve y =
x
x+1
at x = 0.
B) Find an equation of the tangent line to y = 2ex at x = 0 and another tangent line
at x = ln(2).
Problem 8. Suppose that f (2) = 3, g(2) = 2, f 0 (2) = −2,, g 0 (2) = 4 and f 0 (16) = 0.
For the following functions, find h0 (2).
1. h(x) = 5f (x) + 2g(x)
2. h(x) = f (x)g(x)
g(x)
3. h(x) =
1 + f (x)
p
4. h(x) = [f (x)]2 + 7.
5. h(x) = f (x3 · g(x)).
Problem 9. Let f (x) = (3x − 1)ex . For which x is the slope of the tangent line to f
positive? Negative? Zero?
Problem 10. A peg on the end of a rotating crank slides freely in the vertical guide
shown in the diagram. The guide is rigidly connected to a piston which moves horizontally. The crank rotates at a constant angular speed ω in a circle of radius r.
It is easy to show that the position of the
piston is given by
P (t) = L + r cos(ωt),
since x(t) = r cos(ωt). Find the velocity
d
and acceleration of the piston: P (t) and
dt
d2
P (t).
dt2
Problem 11. If a fish is swimming at speed v against a current with speed c (c < v)
the the amount of energy required to swim a fixed distance L is given by
E(v) =
aLv 2
v−c
Find the rate change of Energy with respect to speed v.
Problem 12.
Let
(
t2 + b if t ≤ 0
g(t) =
.
at + 4 if t > 0
Find a and b so that g is differentiable at t = 0.
Math 171 — Calculus I Spring, 2016
Practice Questions for Exam II
Problem 13. By Faraday’s Law, if a conducting wire of length l meters moves at a
velocity v m/s perpendicular to a magnetic field of strength B (in teslas), a voltage is
size E = −Blv is induced in the wire.
A) Calculate
dE
dv
B) Suppose that the wire is oscillating in such a way the v = 10 sin(ωt) where ω is a
positive constant. Find the rate of change of E with respect to time t.
Problem 14.
The figure shows the height y of a mass oscillating at the end of s
spring, through one cycle of the oscillation. Sketch the graph of the velocity as a function
of time.
The figure below shows f , f 0 and f 00 . Determine which is which.
Problem 15.
Problem 16.
Find a formula for the n th derivative of f (x) = e−2x .
Problem 17. Find the equation of the line tangent to the curve y = A tan(ωx) at
π
x=
. Here A and ω are constants.
3ω
Problem 18.
Find the derivative of y with respect to x:
1. e(xy) = cos(y 4 ).
2
2
2
2. x 3 + y 3 = π 3 .
Problem 19. Find an equation of the tangent line to the curve
x2 + sin(y) = xy 2 + 1
at the point (1, 0).
3
4
Problem 20.
equation:
The Lambert function, W (x), is implicitly defined by the following
W (x)eW (x) = x.
Use implicit differentiation to find a formula for
Problem 21.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
d
W (x).
dx
Differentiate each of the following functions:
√
x
f (t) = cos(4πt3 ) + sin(3t + 2)
13. f (x) = x
e
1
g(u) =
ex
cos(u)
14. f (x) = 3
2x
r(θ) = θ3 sin(θ)
x−1
3
x
15. f (x) = (x + 2x + e ) √
s(t) = tan(t) + sec(t)
x
ex
ax + b
g(x) =
16. f (x) =
tan(x)
cx + d
e−2x
x4
h(x) =
17.
f
(x)
=
−2x
1+e
x4 + 2 4
−kt
x(t) = Ae cos(ωt)
18. r(t) = (t − 3)(2t + 1)(t + 5)
√
3
f (x) = 2x3 + 7x + 3
19. y = sin(cos(x))
9
xe2x
f (x) = x8
20.
f
(x)
=
4
x2 + 1
x
2
100
h(x) = 3e + x + 1
21. g(r) = r3 − 1
A
k(x) = 4 + Bx2 + Cx + D
22. h(t) = tan−1 (t2 )
x
2
23. k(z) = sec2 (z + 2)
1
l(x) = x +
24. y = x2 arcsin(x)
x