PRACTICE MIDTERM
Problem
Find the limit ∈ [−∞, ∞], or write “NL” if it doesn’t exist.
e−x
2 −e−x
x
x→∞
(a) lim
ln x
x→∞ 2+sin x
(b) lim
=
=
x
=
(c) lim ln(x+r)−ln
r
r→0
(d) lim
w→0
(e) lim
x→∞
ew −1
w3
ln ln x
x
=
=
Problem
Calculate
R6
(a) 2 x−1/4 dx =
(b)
Rc
(c)
R
(2h + B)dh
(d)
R
( x12 + x5/3 + 5x5 − x20 )dx =
a
3dr
(a, c are constants) =
(B is a constant) =
√
tan( 5s2 − s)ds, (a is a constant) =
(e)
d
dx
Rx
(f)
d
dx
R x2
(g)
d
dx
(h)
2
d
(2t )
dt
a
x
√
cos( t)dt =
sin−1 (2x + 3y)
(y is a constant) =
=
(i) Write as a single integral:
R2
8
f (s)ds −
R 10
8
f (s)ds =
Problem
(a) If g(−1) = 2, g 0 (−1) = 8, find
d −1
g (x)|x=2 .
dx
(b) Given f (x) = x3 + x + 1, find
d
(f −1 (x))|x=1 .
dx
Problem
(a) Write
R9
7
e3x dx as a limit of Riemann sums.
(b) Write the following limit as a definite integral:
n
X
i
i
(5 + )2
lim
n→∞
4n
4n
i=1
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.)
(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)?
(b) What is the average of P (t) on t ∈ [2, 10]?
(c) What is the area under P (t) from t = 2 to t = 10?
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).
(b) Are there any local maxima or minima on the interval (−5, 5), and if so, what are they?
(c) What is the absolute maximum of f (x) on [−5, 5]?
(d) What is the absolute minimum of f (x) on [−5, 5]?
(e) Find all the inflection points of f (x) on (−5, 5).
(f) On what interval(s) is f (x) concave up?
(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 velocity, v(t), on t ∈ (0, 3)?