Practice for Exam 2

Math 1220: Calculus 1
Practice for Exam #2
Exam Thursday, April 2
First of all I’ll give a brief overview of what we have accomplished since the
first exam.
1. We’ve been honing our derivative skills: I expect you to be ninjas with
the chain rule, product rule, etc.
2. We’ve used these rules to expand our library of ‘common knowledge’
derivatives to functions such as ln(x), tan(x), and more (and we’ll do
more in the coming week).
3. We’ve used derivatives to sketch the graphs of functions, accounting
for increasing, decreasing, curving up and curving down behavior.
4. We’ve used derivatives to solve interesting word problems about rates of
change (related rates problems) and optimization (max/min problems).
This is covered in the book sections: §3.10–3.11, §4.1–4.4 and §4.6.
Note Well: In addition to all this, you’ll be expected to draw qualitative slope
and area graphs on the exam.
Book Problems for practice:
§3.10 # 21, 26, 27, 29, 32, 33, 36, 40
§3.11 # 2, 4, 12, 15, 17
4.1 # 37, 39, 40, 47, 48, 52, 55, 56, 59, 60, 62, 66, 79
§4.3 # 23, 28, 32, 35, 38, 42, 43, 50, 51, 58, 62
§4.4 # 15, 16, 19, 27, 28, 33, 34, 43
TWO practice exams follow:
Math 1220: Calculus 1
Practice Exam #2
Problem 1
(a) For the following graph, sketch a qualitative slope and a qualitative area graph. Make sure to label all important points on your
graph.
(b) For the following graph, sketch a qualitative area graph. Make
sure to label all important points on your graph.
Problem 2 Here is a problem that I do NOT WANT YOU TO SOLVE:
A light is moving horizontally at height 30 feet, while a ball
is dropping to the ground. At a certain time,
• the ball is 6 feet above the ground, falling at 2 feet per second
• the light is 18 feet to the right of the ball, moving to the
right at 3 feet per second.
How quickly, and in what direction, is the shadow of the ball
moving at this time?
READ THE PROBLEM THEN DO THIS:
(a) Turn this into a calculus problem. Make sure to be clear about
what your variables mean, and why the calculus problem you write
down will actually solve the word problem given to you.
(b) Explain the strategy for solving your calculus problem. What
steps would you perform? Why would those steps lead to a solution of the problem?
Problem 3 Here is a problem that I do NOT WANT YOU TO SOLVE:
When you look at something, its apparent size depends on the
angle that your eye turns through as you look from one end to
the other. This is why nearby small objects can appear larger
than distant large ones.
Now here is the situation: you are positioned 5 meters from
the street, watching a parade go by. A float goes by at 2 meters
per second, and its platform is exactly at your eye level. Part
of the float is a large cartoon character bobbing up and down so
that its height above the platform at time t is 3 − sin(t) meters,
where t = 0 is the time when it is right in front of you.
At what time does the character appear largest?
READ THE PROBLEM THEN DO THIS:
(a) Turn this into a calculus problem. Make sure to be clear about
what your variables mean, and why the calculus problem you write
down will actually solve the word problem given to you.
(b) Explain the strategy for solving your calculus problem. What
steps would you perform? Why would those steps lead to a solution of the problem?
Problem 4 ( ACTUALLY SOLVE THIS PROBLEM) You’re going
to print a poster which requires 100 square inches of printed area, 2 inch
side margins a 3 inch top margin and 5 inch bottom margin. What are the
dimensions of a poster which will have the smallest total area?1 Make sure
to explain your work.
1
Bonus! Instead solve the generic problem which requires printed area P , side margins
l and r and top and bottom margins t and b.
Problem 5 Here is a problem that I do NOT WANT YOU TO SOLVE:
The amount of fun that you have doing something is given by
the simple formula
Fun = (difficulty) · (level of success)
Success and difficulty (which I’ll denote s and d, respectively) are
related to each other by the formula
(level of success) =
(time spent)
.
1 + (difficulty)2
One afternoon, you find yourself engaged in a bunch of different
activities, so that at time t o’clock, the difficulty level is d(t) =
5t cos(t2 ). How much fun do you have between 1 and 4?
READ THE PROBLEM THEN DO THIS:
(a) Turn this into a calculus problem. Make sure to be clear about
what your variables mean, and why the calculus problem you write
down will actually solve the word problem given to you.
(b) Explain the strategy for solving your calculus problem. What
steps would you perform? Why would those steps lead to a solution of the problem?
Math 1220: Calculus 1
Another Practice Exam!
Problem 6
(a) For the following graph, sketch a qualitative slope graph. Make
sure to label all important points on your graph.
(b) For the following graph, sketch a qualitative area graph. Make
sure to label all important points on your graph.
Problem 7 (ACTUALLY SOLVE THIS PROBLEM) You are on a
Ferris wheel with radius 35 feet, looking wistfully at the funnel cake stand
that is in line with the wheel, but a 100 feet distant from where it comes to
ground level. The wheel is turning at one revolution every minute. When
you are at the very top, how quickly is the distance between you and sweet
sweet funnel cake decreasing?
Problem 8 Here is some data about the signs of the derivatives of a function
f . Use the information to sketch the graph of f , being careful to mark the
important points such as maxima, minima, inflection, etc.
Problem 9 (ACTUALLY SOLVE THIS PROBLEM) Consider the
region under the parabola y = 4 − x2 and over the x-axis. You want to put
a rectangle in this region with the greatest possible area. Which rectangle is
it?
Hint You may take for granted that one side of the rectangle is on the x-axis,
and that the other two corners are on the parabola.
Problem 10 Use a linear approximation to estimate (2.2)2 . How much
accuracy can you guarantee? Explain your answers and show your work.