PRACTICE MIDTERM
Problem
Find the limit ∈ [−∞, ∞], or write “NL” if it doesn’t exist.
e−x
2 −e−x
x
x→∞
(a) lim
=
0
ln x
x→∞ 2+sin x
(b) lim
=
∞
x
(c) lim ln(x+r)−ln
=
r
r→0
1/x
(d) lim
w→0
ew −1
w3
=
∞
(e) lim
x→∞
ln ln x
x
=
0
Problem
Calculate
R6
(a) 2 x−1/4 dx =
4/3(63/4 − 23/4 )
(b)
Rc
a
3dr
(a, c are constants) =
3(c − a)
(c)
R
(2h + B)dh
(B is a constant) =
2h / ln 2 + Bh + C
(d)
R
( x12 + x5/3 + 5x5 − x20 )dx =
−1/x + 3/8x8/3 + 5/6x6 − x2 1/21 + C
(e)
d
dx
Rx
a
√
tan( 5s2 − s)ds, (a is a constant) =
tan
(f)
d
dx
R x2
x
√
5x2 − x
√
cos( t)dt =
√
2x cos x − cos x
(g)
d
dx
sin (2x + 3y)
−1
(y is a constant) =
√
(h)
2
d
(2t )
dt
2
1−(2x+3y)2
=
2
2t(ln 2)2t
(i) Write as a single integral:
R2
8
f (s)ds −
R 10
8
f (s)ds =
−
R 10
2
f (s)ds
Problem
(a) If g(−1) = 2, g 0 (−1) = 8, find
d −1
g (x)|x=2 .
dx
1/8
(b) Given f (x) = x3 + x + 1, find
d
(f −1 (x))|x=1 .
dx
1
f 0 (0)
=1
Problem
(a) Write
R9
7
e3x dx as a limit of Riemann sums.
n
P
2 3(7+2i/n)
e
n→∞ i=1 n
lim
(b) Write the following limit as a definite integral:
n
X
i
1
lim
(5 + )2
n→∞
4n
4n
i=1
R 5+1/4
5
x2 dx
Problem
Suppose you are trying to use Newton’s method to find a positive value of x that solves f (x) = 0,
where f (x) = e−x + sin x.
(a) Write down an expression for the next best guess x2 , if you start with an initial guess of
x1 = 1. (Don’t evaluate it.)
1−
e−1 +sin 1
−e−1 +cos 1
(b) Describe one thing that could go wrong with Newton’s method.
Problem
The rate of growth of a population is given by
dP
dt
= 0.5P (t).
(a) If the population at time t = 0 is 1000, what is P (t)?
1000e0.5t
(b) What is the average of P (t) on t ∈ [2, 10]?
250(e5 − e)
(c) What is the area under P (t) from t = 2 to t = 10?
2000(e5 − e)
Problem 6
Consider the function f (x) = sin x − x on the interval −5 ≤ x ≤ 5.
(a) Find the critical points in (−5, 5) (you don’t have to consider the endpoints).
x=0
(b) Are there any local maxima or minima on the interval (−5, 5), and if so, what are they?
no (since f 0 (x) = cos x − 1 is always < 0)
(c) What is the absolute maximum of f (x) on [−5, 5]?
f (−5) = − sin 5 + 5
(d) What is the absolute minimum of f (x) on [−5, 5]?
f (5) = sin 5 − 5
(e) Find all the inflection points of f (x) on (−5, 5).
x = 0, ±π
(f) On what interval(s) is f (x) concave up?
x ∈ (−π, 0), (π, 5)
(g) Sketch a graph of f (x). It doesn’t have to be completely precise, but make sure you have the
right local maxima/minima and that it is concave up / concave down in the right places.
Problem 6
(a) State the Mean Value Theorem for derivatives.
(b) Suppose a plane flies 300km in 3 hrs. What does the Mean Value Theorem tell you about the
slope of at least one of the tangent lines of its position, x(t), on t ∈ (0, 3)?
There is at least one value of t in (0, 3), say t = c, where the slope of the tangent line to x(t)
equals 300/3: x0 (c) = 100.