Math 1A: Introduction to functions and calculus Oliver Knill, Spring 2020

5/7/2020: Final Practice E

Your Name:

• Solutions are submitted as PDF handwritten in a file called after your name.
Capitalize the first letters like OliverKnill.pdf. The paper has to feature your
personal handwriting and contain no typed part. If you like, you can start
writing on a new paper. For 1), you could write 1: False, 2: False · · · 20: False.
Sign your paper.
• No books, calculators, computers, or other electronic aids are allowed. You can
use one page of your own handwritten notes when writing the paper. It is your
responsibility to submit the paper on time and get within that time also a con-
firmation.

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Total: 140
Problem 1) TF questions (20 points) No justifications are needed.
If a function f (x) has a critical point 0 and f 00 (0) = 0 then 0 is neither a
1) T F maximum nor minimum.
Rx
2) T F If f 0 = g then 0 g(x) = f (x).

3) T F The function f (x) = 1/x has the derivative log(x).

4) T F The function f (x) = arctan(x) has the derivative 1/ cos2 (x).

Rb
The fundamental theorem of calculus implies that a f 0 (x) dx = f (b) −
5) T F f (a).
6) T F limx→8 1/(x − 8) = ∞ implies limx→3 1/(x − 3) = ω.

A continuous function which satisfies limx→−∞ f (x) = 3 and limx→∞ f (x) =

7) T F 5 has a root.
8) T F The function f (x) = (x7 − 1)/(x − 1) has a limit at x = 1.

If fc (x) is an even function with parameter c and f 0 (0) = 0 and for c < 3
9) T F the function is concave up at x = 0 and for c > 3 the function is concave
down at x = 0, then c = 3 is a catastrophe.
√
10) T F The function f (x) = + x2 has a continuous derivative 1 everywhere.
A rower rows on the Charles river leaving at 5 PM at the Harvard boat
11) T F house and returning at 6 PM. If f (t) is the distance of the rower at time t
to the boat house, then there is a point where f 0 (t) = 0.
12) T F A global maximum of a function f (x) on the interval [0, 1] is a critical point.

A continuous function on the interval [2, 3] has a global maximum and global
13) T F minimum.
The intermediate value theorem assures that if f is continuous on [a, b] then
14) T F there is a root of f in (a, b).
On an arbitrary floor, a square table can be turned so that it does not
15) T F wobble any more.
16) T F The derivative of log(x) is 1/x.
Rx
If f is the marginal cost and F = 0 f (x) dx the total cost and g(x) =
17) T F F (x)/x the average cost, then points where f = g are called ”break even
points”.
At a function party, Log talks to Tan and the couple Sin and Cos, when she sees
her friend Exp alone in a corner. Log: ”What’s wrong?” Exp: ”I feel so lonely!”
18) T F Log: ”Go integrate yourself!” Exp sobbs: ”Won’t change anything.” Log: ”You
are so right”.
If a car’s position at time t is f (t) = t3 − t, then its acceleration at t = 1 is
19) T F 6.
 2du
For trig substitution, the identities u = tan(x/2), dx = (1+u2 )
, sin(x) =
20) T F 2u 1−u2
1+u2
, cos(x) = 1+u2
are useful.

Problem 2) Matching problem (10 points) No justifications are needed.

a) Match the following integrals with the graphs and (possibly signed) areas.

Integral Enter 1-6 Integral Enter 1-6

R1 3
R1
sin(πx)x dx. (1 + sin(πx)) dx.
R −1
1
−1R
1
−1
log(x + 2) dx. sin2 (x) dx.
R1 R−1
1
−1
x + 1 dx. −1
x2 + 1 dx.

1) 2) 3)

4) 5) 6)

Problem 3) Matching problem (10 points) No justifications are needed.

Determine from each of the following functions, whether discontinuities appears at

x = 0 and if, which of the three type of discontinuities it is at 0.
 Function Jump discontinuity Infinity Oscillation No discontinuity
5
f (x) = log(|x| )
f (x) = cos(5/x))
f (x) = cot(1/x)
f (x) = sin(x2 )/x3
f (x) = arctan(tan(x − π/2))
f (x) = 1/ tan(x)
f (x) = 1/ sin(x)
f (x) = 1/ sin(1/x)
f (x) = sin(exp(x))/ cos(x)
f (x) = 1/ log |x|

Problem 4) Area computation (10 points)

Find the area of the region enclosed by the graphs of the function f (x) = x4 − 2x2 and
the function g(x) = −x2 .

Problem 5) Volume computation (10 points)

A farmer builds a bath tub for his warthog ”Tuk”. The bath has triangular shape of
length 10 for which the width is 2z at height z. so that when filled with height z the
surface area of the water is 20z. If the bath has height 1, what is its volume?

P.S. Don’t ask how comfortable it is to soak in a bath tub with that geometry. The
answer most likely would be ”Noink Muink”.
Problem 6) Definite integrals (10 points)

Find the following definite integrals

R2√ √
a) (3 points) 1
x + x2 − 1/ x + 1/x dx.
R2 √
b) (3 points) 1
2x x2 − 1 dx
R2
c) (4 points) 1
2/(5x − 1) dx

Problem 7) Anti derivatives (10 points)

Find the following anti-derivatives

3
R
a) (3 points) 1+x2
+ x2 dx
 R tan2 (x)
b) (3 points) cos2 (x)
dx
R
c) (4 points) log(5x) dx.

Problem 8) Chain rule (10 points)

A juice container of volume V = πr2 h changes radius r but keeps the height h = 2
fixed. Liquid leaves at a constant rate V 0 (t) = −1. At which rate does the radius of
the bag shrink when r = 1/2? Differentiate V (t) = πr(t)2 h for the unknown function
r(t) and solve for r0 (t), then evaluate for r = 1/2.
Problem 9) Global extrema (10 points)

We build a chocolate box which has 4 cubical containers

of dimension x × x × h. The total material is f (x, h) =
4x2 + 12xh and the volume is 4x2 h. Assume the volume
is 4, what geometry produces the minimal cost?

Problem 10) Integration techniques (10 points)

Which integration technique works? It is enough to get the right technique and give
the first step, not do the actual integration:
R
a) (2 points) (x2 + x + 1) sin(x) dx.
R
b) (2 points) x/(1 + x2 ) dx.
R√
c) (2 points) 4 − x2 dx.
R
d) (2 points) sin(log(x))/x.

1
R
e) (2 points) (x−6)(x−7)
dx.

Problem 11) Hopital’s rule (10 points)

Find the following limits as x → 0 or state that the limit does not exist.
tan(x)
a) (2 points) x
 x
b) (2 points) cos(x)−x
.

c) (2 points) x log(1 + x)/ sin(x).

d) (2 points) x log(x).

e) (2 points) x/(1 − exp(x)).

Problem 12) Applications (10 points)

The cumulative distribution function on [0, 1]

2 √
F (x) = arcsin( x)
π
defines the arc-sin distribution.

a) Find the probability density function f (x) on [0, 1].

R1
b) Verify that 0 f (x) dx = 1.
Remark. The arc sin distribution is important chaos theory and probability theory.
Problem 13) Data (10 points)

Find the best linear fit y = mx through the data points (3, 5), (1, 1), (−1, 1), (2, 2).