MATH 2413 Houston Community College System Algebra Questionnaire

Math 2413Homework Assignment 8 (Written)
Applications of the Derivative (Chapter 3)
Full Name:
PSID:
Instructions
• Print out this file, fill in your name and ID above, and complete the problems. If the problem is from the text, the
section number and problem number are in parentheses. (Note: if you cannot print out this document, take the time
to carefully write out each problem on your own paper and complete your work there.)
• Use a blue or black pen (or a pencil that writes darkly) so that when your work is scanned it is visible in the saved
image.
• Write your solutions in the spaces provided. Unless otherwise specified, you must show work in order receive credit for
a problem.
• Students should show work in the spaces provided and place answers in blanks when provided, otherwise BOX final
answer for full credit.
• Once completed, students need to scan a copy of their work and answers and upload the saved document as a .pdf
through CASA CourseWare.
• This assignment is copyright-protected; it is illegal to reproduce or share this assignment (or any question from it)
without explicit permission from its author(s).
• No late assignments accepted.
1. (10 points) Fun-loving Professor Caglar loves standing on top of a 230 ft building and tossing water balloons at
unsuspecting people below, especially on Professor Mosas Sosa. If she drops a balloon, how long will it take for it to hit
Moses (who is 5 feet tall)?
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2. (12 points) As most UH Cougars know, our main campus is overrun with squirrels, some of whom actively dash
around and others that sit around waiting for students and faculty to feed them. The graph below display’s one particular
squirrel’s running activity on a recent day, with the vertical y-axis keeping track of its North-Soth position (in feet) relative
to the Library, and the x-axis keeping track of time (in minutes).
Using two or three sentences, write a short story about the squirrel’s travels that match the graph. Be sure to include the
phrases “sat still on a bench,” “returned to the library,” “ran north,” and “ran south.”
2
3. (12 points) Engineering students at the University of Houston can take classes where they learn about the theory
of electrical circuits. One topic they encounter in these courses is called “Ohm’s law,” and it describes the relationship
between the voltage V across a resistor, the electrical current I passing through the resistor, and a quantity R known as the
resistance. The law can be written as follows:
V = IR.
Voltage is typically measured in volts, while current is measured in amperes (amps), and resistance is measured in ohms
where 1 ohm equals 1 volt/amp. Lastly, in circuits with variable resistance, the quantities V, I, and R can depend on time.
(a) (4 points) Differentiate Ohm’s Law to find an equation relating the quantities V, I, R,
dR
dV dI
,
and
.
dt dt
dt
(b) (4 points) Suppose that in an electrical circuit the current is increasing at a rate of 0.5 amps per second and the
resistance is decreasing at a rate of 4 ohms per second. If at this same moment in time there are 3 volts of voltage and
2 ohms of resistance, what is the rate of change in voltage at this time? Is the voltage increasing or decreasing?
(c) (4 points) Suppose that in an electrical circuit the current is increasing at a rate of 0.5 amps per second and the voltage
is decreasing at a rate of 2 volts per second. If at this same moment in time there are 3 volts of voltage and 2 ohms of
resistance, what is the rate of change in resistance at this time? Is the resistance increasing or decreasing?
3
4. (12 points) Answer the following questions about the function g(t) = −
(a) (1 point) The domain of g(t) is
2
.
1 + t2
(no work need be shown for this part).
(b) (5 points) On which intervals is g(t) increasing and on which intervals is g(t) decreasing?
(c) (6 points) On which intervals is g(t) concave up and on which intervals is g(t) concave down?
4
3
5. (10 points) Use Calculus to find the absolute and local extreme values of f (x) = x + x2/3 on the interval [−8, 8].
2
6. (10 points) Find the x-coordinates of the inflection points for the polynomial p(x) =
5
x5
5×4
2022
−
+
.
20
12
π
7. (6 points) Given the function g(t) = t2 + 3t + 12 on the interval [0, 1], find all values of c ∈ (0, 1) at which the
conclusion of The Mean Value Theorem is satisfied.
8. (8 points) Sketch a graph of a differentiable function y = f (x) (whose domain is the entire real line) that satisfies the following properties (no work need be shown for this problem):
(a) (2 points) f (x) is increasing on (−∞, −1).
(b) (2 points) f (x) is decreasing on (−1, ∞).
(c) (2 points) f (x) is concave down on (−∞, 0).
(d) (2 points) f (x) is concave up on (0, ∞).
4
3
2
1
-4
-3
-2
-1
1
-1
-2
-3
-4
6
2
3
4
9. (15 points) Sketch a graph of the function f (x) =
points of inflection.
x−3
, showing all asymptotes, intercepts, extrema and
x2 − 5x + 6
4
3
2
1
-2
-1
1
2
-1
-2
-3
-4
7
3
4
5
6
10. (5 points) One topic I studied / am studying in preparation for Test 3 is …
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