MATH 2413
HOMEWORK ASSIGNMENT 8 (WRITTEN)
APPLICATIONS OF THE DERIVATIVE (CHAPTER 3)
Full Name:
PSID:
INSTRUCTIONS
F Print out this file, fill in your name and ID above, and complete the problems. If the problem is from the text, the
o 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.)
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(F 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
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F5 Write your solutions in the spaces provided. Unless otherwise specified, you must show work in order receive credit for
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Students should show work in the spaces provided and place answers in blanks when provided, otherwise BOX final
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Once completed, students need to scan a copy of their work and answers and upload the saved document as a .pdf
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This assignment is copyright-protected; it is illegal to reproduce or share this assignment (or any question from it)
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(10 points) Fun-loving Professor Caglar loves standing on top of a 230 ft building and tossing water balloons at
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unsuspecting
people below, especially on Professor Mosas Sosa. If she drops a balloon, how long will it take for it to hit
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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.”
After covering a 4ft distance away from the library and running for 2 minutes with an average speed of 2ft/minutes, the squirrel
decides to rest and sat still on a bench for 3 minutes. Upon regaining its strength, it runs at a speed of 4ft/second for another 1 minutes
towards its original position and has returned to the library after covering a distance of 8ft. Upon realizing it returned to its original
position, it ran north for 2 minutes covering a distance of 2 ft with a speed of 1 ft/minute but then decides to run south passing the
library after 2 minutes with an average speed of 2ft/minute and stopping 2ft south from the library.
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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,
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dV dI and dR .
,
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?
Using the above formula and knowing that;
(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?
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4
4. (12 points) Answer the following questions about the function g(t) = −
(a) (1 point) The domain of g(t) is (-∞,∞)
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?
To find when a function is concave, you must first take the 2nd derivative, then set it equal to 0 and solve for t, and then find between
which zero values the function is negative
Test values on all sides of these to find when the function is negative, and therefore decreasing. I will test the values of -1, 0.4, and -1.
5
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5. (10 points) Use Calculus to find the absolute and local extreme values of f (x) = x + x 2/3 on the interval [−8, 8].
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Step 1: Find all critical values of f on the interval [−8, 8].
Step 2: Evaluate f at the critical values from Step 1 and at the endpoints -8 and 8.
Step 3: Choose the largest value as the absolute maximum value,and choose the smallest value as the absolute minimum value.
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6. (10 points) Find the x-coordinates of the inflection points for the polynomial p(x) =
Substitute a value from the interval (−∞,0), (0,5), (5, ∞) Into
the second derivative to determine if it is increasing or
decreasing.
Find the second derivative
*First derivative
𝑓 ′ (𝑥) =
f’’(x)=x^3 – 5x^2
x5
−
20
x 4 5𝑥^3
−
4
3
For (−∞,0):
Replace the variable x with −0.1 in the expression.
f''(−0.1)=(−0.1)3−5(−0.1)2
The final answer is
0.051
A. Set the second derivative equal to 0
Then solve the equation
X3−5X2=0.
Set the second derivative equal to 0 then solve for x
X^3−5X^2=0
X^2(X−5)=0 Hence
x = 0 and x = 5
At −0.1-, the second derivative is − 0.051. Since this is
negative, the second derivative is decreasing on
the interval (−∞,0).
Decreasing on (−∞,0)(-∞,0) since f''(x)<0
Split (−∞,∞) Into intervals around the points that could
potentially be inflection points.(−∞,0)∪(0,5)∪(5
(−∞,∞)(-∞,∞) into intervals around the points that
could potentially be inflection points.
For (0,5)
(−∞,0)∪(0,5)∪(5,∞)(-∞,0)∪(0,5)∪(5,∞)
Replace the variable x with 5/2 In the expression.
f‘’(5/2)=(5/2)^
−5(5/2)^2
Substitute
a value
from the interval (−∞,0)(5
5
5
𝑓" to
= (determine
) = ( )^3if−it (is )^2
∞,0) into the second derivative
2
2
2
f”=−125/8 or decreasing.
increasing
Replace the variable
X With 52 In the expression.
The final answer is
At 5/2, the second derivative is 15.625. Since this is negative, the second derivative is
decreasing on the interval
Decreasing on (0,5) Since ‘’f(X)<0
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For (5,∞).Replace the variable X With 5.1 in the expression. f‘’(5.1)=(5.1)^3−5(5.1)^2.
f‘’(5.1)=132.651 – 130.05
f‘’(5.1)=2.601
At 5.1 , the second derivative is 2.601. Since this is positive, the second derivative is
increasing on the interval (5,∞). Increasing on (5,∞) Since f‘’()>0
An inflection point is a point on a curve at which the concavity changes sign
from plus to minus or from minus to plus.
The x-coordinate of the inflection point in this case is x=5
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7. (6 points) Given the function g(t) = t2 + 3t + 12 on the interval [0, 1], find all values of c S (0, 1) at which the
conclusion of The Mean Value Theorem is satisfied.
u
b
s
ti
t and differentiable
The function is a polynomial which is continuous and differentiable everywhere and so will be continuous on [0,1]
u
on (0,1).
t
Therefore, the conditions for the Mean Value Theorem are met and so we can actually do the problem.
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Now that we know that the Mean Value Theorem can be used there really isn’t much to do. All we need to do isvdo some function
evaluations and take the derivative
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t(0)= 12
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t(1)=16
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t’= 2t + 3
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Plug into the formula
from the Mean Value Theorem and solve for c.
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Solution:
Verify first if the Mean Value Theorem can be used in the given
16−12
1−0
=2c + 3
(replace t with c)
-2c + 4 -3 =0
c=
1
2
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−
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,
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),
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0
,
5
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5
,
∞
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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, ∞).
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9. (15 points)
x−3
Sketch a graph of the function f (x) =
x2
points of inflection.
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2
Intercepts:
y-intercept : (0, -1/2
x- intercept : none
1
2
, showing all asymptotes, intercepts, extrema and
Extremes points:
none
Inflection point: none
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-
− 5x + 6
-1
1
2
3
4
-1
ASYMPTOTES
Vertical : x= 2
Horizontal: y = 0
-2
v
-3
-4
1
5
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10. (5 points) One topic I studied / am studying in preparation for Test 3 is ...
Application of derivatives including curve sketching, differentiation, optimization and related rates under
numerous conditions.
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