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MAT250_TW_V3 | Chapter 8: Curve Sketching
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9/5/2021 Graded Midterm Exam
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MAT250_TW_V3 | Chapter 8: Curve Sketching
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Started on Sunday, September 5, 2021, 2:04 AM
State Finished
Completed on Sunday, September 5, 2021, 3:47 AM
Time taken 1 hour 43 mins
Grade 160.00 out of 200.00 (80%)
Suppose a particle’s position is given by , where is measured in
seconds and is given in centimeters. What is the velocity of the particle when
?
Select one:
a.
b.
c.
d.
On which interval is increasing?
Select one:
a.
b. and
c.
d.
f(t) = t4 t
f(t)
t = 3
v = 81 cm/sec
v = 108 cm/sec
v = 324 cm/sec
v = 1728 cm/sec
f (x) = x 3 −3x2 − 9x + 1
(−1, 3)
(−∞, −1) (3, ∞)
(−∞, 1)
(1, ∞)9/5/2021 Graded Midterm Exam
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Consider the function
Is continuous at ?
Select one:
a. Yes, is continuous at .
b. No, f(x) is not continuous at x = 0.
c. There is not enough information.
Use the linear approximation method to estimate the value of \( \ln(2.7^2) \) in
terms of \( e \).
Select one:
a. \( \displaystyle \frac{5.4}{e} \)
b. \( \displaystyle \frac{5.4+e}{e} \)
c. \( 4- \displaystyle\frac{5.4}{e} \)
d. \( \displaystyle \frac{2e-5.4}{e^2} \)
What is the slope of the line tangent to the curve \( x^2y^2 = 1 \) at the point \( (1,
1) \)?
Select one:
a. \(0\)
b. \(1\)
c. The slope is unde¦ned.
d. \(-1\)
f(x) = x3 + x
x
f(x) x = 0
f(x) x = 09/5/2021 Graded Midterm Exam
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Question 7
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Question 8
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What is the derivative of the function \( f(x) = x^3 +2e^{2x} \)?
Select one:
a. \( 3x^2+4e^{2x} \)
b. \( 3x^2+2e^{2x} \)
c. \( 3x^2+2e^2 \)
d. \( 3x^2+2e^x \)
Use implicit differentiation to ¦nd an equation of the line tangent to the curve \( x^2
+ y^2 = 10 \) at the point \( (3, 1) \).
Select one:
a. \(y=-x\)
b. \(y=x\)
c. \(y=-3x+10\)
d. \(y=3x-8\)
A ball is rolling along the \( x \)-axis. Its position (in feet) at time \( x \) (in seconds)
is given by \( f(x) = 2\sqrt{x} \). Find its instantaneous rate of change when \( x=9
\) seconds.
Select one:
a. \( 6 \, \hbox{ft/sec} \)
b. \( \displaystyle\frac{1}{12} \, \hbox{ft/sec} \)
c. \( \displaystyle\frac{1}{3}\, \hbox{ft/sec} \)
d. \( \displaystyle\frac{2}{3} \, \hbox{ft/sec} \)9/5/2021 Graded Midterm Exam
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Question 9
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Sketch the parabola of equation \( y=x^2-6x+9 \), and indicate its vertex.
Select one:
a.
b.
c.
d.9/5/2021 Graded Midterm Exam
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What is the derivative of the function
\( f(x) = -\, \displaystyle \frac{1}{4x^6} \)?
Select one:
a. \( \displaystyle \frac{3}{2x^7} \)
b. \( -\, \displaystyle \frac{3}{2x^5} \)
c. \( \displaystyle \frac{3}{2x^5} \)
d. \( -\, \displaystyle \frac{3}{2x^7} \)
If \( \displaystyle\frac{dy}{dx}= 0 \) for a given value of \( x \), then the line tangent
to the curve \( y = f (x) \) at that value is horizontal.
Select one:
a. true
b. false9/5/2021 Graded Midterm Exam
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Question 13
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What is the limit of the function in the graph at \( x=4 \)?
Select one:
a. \( \infty \)
b. \( 6 \)
c. The limit does not exist.
d. \( 2 \)
Consider the function \( y=x^2 + x + 9 \). What is the equation of the tangent line at
\( x = 2 \)?
Select one:
a. \(y=-5x+5\)
b. \(y=5x+5\)
c. \(y=5x-5\)
d. \(y=-5x-5\)9/5/2021 Graded Midterm Exam
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Question 15
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Question 16
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The position of a car at time \( t \) is given by the function \( p (t) = t^2 + 4t − 17 \).
Where will the car be when it moves at a velocity of \(10\)? Assume \( t \geq 0 \).
Select one:
a. \(-4\)
b. \(4\)
c. \(3\)
d. \(-3\)
A man \(6\) feet tall is walking toward a building at the rate of \(5\;\hbox{ft/sec}\).
If there is a light on the ground \(50\;\hbox{ft}\) from the building, how fast is the
man’s shadow on the building growing shorter when he is \(30\;\hbox{ft}\) from
the building?
Select one:
a. \( - \displaystyle\frac{5}{6} \;\hbox{ft/sec} \)
b. \( -\displaystyle \frac{65}{6}\;\hbox{ft/sec} \)
c. \( -\displaystyle \frac{15}{4}\;\hbox{ft/sec} \)
d. \( -\displaystyle \frac{45}{4}\;\hbox{ft/sec} \)
Compute the derivative of the function \( f(x) = \displaystyle \frac{x-x^2}{3x+5} \).
Select one:
a. \( \displaystyle \frac{(1-2x)(3x+5)\,+\,3(x-x^2)}{(3x+5)^2} \)
b. \( \displaystyle \frac{(1-2x)(3x+5)\,-\,3(x-x^2)}{(3x+5)^2} \)
c. \( \displaystyle \frac{1-2x}{(3x+5)^2} \)
d. \( \displaystyle \frac{1-2x}{3} \)9/5/2021 Graded Midterm Exam
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Question 18
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Given \( xy = 5 \), ¦nd \( \displaystyle\frac{dy }{dx} \) by implicit differentiation.
Select one:
a. \( \displaystyle\frac{y}{x} \)
b. \( \displaystyle\frac{x}{y} \)
c. \( - \displaystyle\frac{y}{x} \)
d. \(0\)
The given region is the combination of a semicircle with radius \( 2 \) cm and a
right isosceles triangle with base length \( 4 \) cm. Find the area of the given
region using the formula for the area of a circle, \( A = \pi r^2 \) where \( r \) is the
circle's radius, and the formula for the area of a triangle, \( A =
\displaystyle\frac{bh}{2} \) where \( b \) and \( h \) are the lengths of the triangle's
base and height, respectively. Remember, a semicircle's area is one half the area of
the circle with the same radius.
Select one:
a. \( 8+2π\,\,cm^2 \)
b. \( 16+4π\,\,cm^2 \)
c. \( 16+2π\,\,cm^2 \)
d. \( 8+4π\,\,cm^2 \)
e. None of the above9/5/2021 Graded Midterm Exam
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Question 20
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Differentiate the function \( f(x) = 2x \).
Select one:
a. \(2x\)
b. \(0\)
c. \(x^2\)
d. \(2\)
Evaluate the following as true or false:
If \( f ''(x) < 0 \) and \( f '(x) = 0 \) at the point \( (x, f(x)) \), then the point must be a
relative maximum.
Select one:
True
False
Given \( xy = y^3 \), ¦nd \( \displaystyle\frac{dy}{dx} \) by implicit differentiation.
Select one:
a. \( \displaystyle\frac{1}{2y} \)
b. \( \displaystyle\frac{y}{3y^2-x} \)
c. \( \displaystyle \frac{y}{x-3y^2} \)
d. \(2y\)9/5/2021 Graded Midterm Exam
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Question 24
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Let \( f'(x) = 3x^2+4x \) de¦ne the instantaneous rate of change (in \
(\hbox{ft/min}\)) of a car moving along the \(x\)-axis. What is the instantaneous
rate of change at time \(1 \, \hbox{min}\)?
Select one:
a. \(10 \, \hbox{ft/min}\)
b. \(7 \, \hbox{ft/min}\)
c. \(3 \, \hbox{ft/min}\)
d. \(4 \, \hbox{ft/min}\)
A man \(6\) feet tall walks at a rate of \(5\;\hbox{ft/sec}\) away from a light that is
\(15\) feet above the ground. When he is \(10\) feet from the base of the light, at
what rate is the length of his shadow changing?
Select one:
a. \( \displaystyle \frac{25}{3}\;\hbox{ft/sec} \)
b. \(\displaystyle \frac{5}{3}\;\hbox{ft/sec} \)
c. \( \displaystyle \frac{10}{3}\;\hbox{ft/sec} \)
d. \(\displaystyle \frac{5}{9}\;\hbox{ft/sec} \)
Compute the derivative of the function \( f (x) = \sec x \).
Select one:
a. \(\cos x\)
b. \(- \tan x \sec x\)
c. \( \sin^2 x\)
d. \( \tan x \sec x\)9/5/2021 Graded Midterm Exam
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Question 27
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Determine, if it exists, \( \displaystyle {\lim_{x→-2}}\,\,\frac{1+\frac{2}{x}}{x-
\frac{4}{x}} \)
Select one:
a. \( \displaystyle \frac{1}{4} \)
b. \( \displaystyle -\frac{1}{4} \)
c. \( 1 \)
d. The limit does not exist.
What is the derivative of the function
\( f (x) =\log _x 2 \)?
Select one:
a. \( -\displaystyle\frac{\ln 2}{x(\ln x)^2} \)
b. \( \displaystyle\frac{2}{x} \)
c. \( 2x \)
d. \( \displaystyle\frac{\ln 2}{x(\ln x)^2} \)
True or false?
Let \(A\) be the area of a circle with radius \(r\) that is changing with respect to
time. If \( \displaystyle\frac{dr}{dt} \) is constant, then \( \displaystyle\frac{dA}{dt}
\) is also constant.
Select one:
True
False9/5/2021 Graded Midterm Exam
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Compute the derivative of the function \( f(x) = \tan(2x+1) \).
Select one:
a. \( \sec^2(2x+1)\)
b. \(2 \sin^2(2x+1)\)
c. \(-2 \sec^2(2x+1)\)
d. \(2 \sec^2(2x+1)\)
Determine \( \displaystyle\frac{d^{2}}{dx^2} [e^{3x}] \)
Select one:
a. \( 3e^{9x} \)
b. \( 9e^{3x} \)
c. \( 3e^{3x} \)
d. \( 9x^2 e^{3x} \)
Which of the following equations could be an equation of a vertical asymptote of
\( y= \displaystyle \frac{x^2-2x}{x^2-6x+8} \)
Select one:
a. \( x=0 \)
b. \( x=1 \)
c. \( x=2 \)
d. \( x=4 \)9/5/2021 Graded Midterm Exam
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Question 32
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On Friday night, Ken drove one and a half hours at ¦fty-¦ve miles an hour to attend
a concert in Austin. How far did he drive?
Select one:
a. \( 36.67 \) miles.
b. \( 82.5 \) miles.
c. \( 0.03 \) miles.
d. \( 56.5 \) miles.
Compute the derivative of the function \( f(x) = \cot(2x-1) \)?
Select one:
a. \( -2 \sin(2x-1) \)
b. \( 2 \csc^2(2x-1) \)
c. \( -2 \csc^2(2x-1) \)
d. \( - \csc^2(2x-1) \)9/5/2021 Graded Midterm Exam
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Question 34
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What is the limit of the function in the graph at \( x=4 \)?
Select one:
a. \( 6 \)
b. \( 2 \)
c. \( 4 \)
d. The limit does not exist.
Compute the derivative of the function
\( f(x) = \displaystyle \frac{2x}{(3x^{2}+2)^{3/2}} \).
Select one:
a. \( -\, \displaystyle \frac{12x^{2}}{(3x^{2}+2)^{3}} \)
b. \( \displaystyle \frac{2(3x^{2}+2)^{3/2}-18x^{2}(3x^{2}+2)^{1/2}}
{(3x^{2}+2)^{3}} \)
c. \( \displaystyle \frac{2(3x^{2}+2)^{3}+36x^{2}}{(3x^{2}+2)^{3}} \)
d. \( \displaystyle \frac{2(3x^{2}+2)^3-36x^2(3x^{2}+2)^{1/2}}{(3x^{2}+2)^{3}} \)9/5/2021 Graded Midterm Exam
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A particular rectangle has a length of \( 3 \) feet and a width of \( 2 \) feet. What is
the area of the described rectangle?
Select one:
a. \( 3 \) square feet.
b. \( 10 \) square feet.
c. \( 5 \) square feet.
d. \(6 \) square feet.
What are the \( x- \)coordinates of the points of in§ection of \( f(x) = x(x-4)^3 \)?
Select one:
a. \( x=1 \) only
b. \( x=2 \) only
c. \( x=1 \) and \( x=4 \)
d. \( x=2 \) and \( x=4 \)9/5/2021 Graded Midterm Exam
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Question 37
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What is the limit of the function in the graph at \( x=4 \)?
Select one:
a. \( 6 \)
b. \( 4 \)
c. \( 8 \)
d. The limit does not exist.9/5/2021 Graded Midterm Exam
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Question 39
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If the graph of the derivative of \( f(x) \) is shown, at which \( x \)-coordinates
would \( f(x) \) have a point of in§ection?
Select one:
a. \( x=q \) only
b. \( x=q \) and \( x=s \)
c. \( x=r \) and \( x=s \)
d. \( x=p \) and \( x=s \)
A body is thrown upward so that its height \( h \) at the end of \( t \) seconds is \(
h=160t-16t^2 \) ft. Which of the following is its maximum height?
Select one:
a. \( 800 \)
b. \( 400 \)
c. \( 200 \)
d. \( 600 \)9/5/2021 Graded Midterm Exam
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Compute the derivative of the function:
\( f(x) = \bigg( \displaystyle \frac{x+1}{x-1} \bigg)^{5} \).
Select one:
a. \( 5 \bigg( \displaystyle \frac{x+1}{x-1} \bigg)^{4} \)
b. \( 10x \displaystyle \frac{(x+1)^{4}}{(x-1)^{6}} \)
c. \( -10 \displaystyle \frac{(x+1)^{4}}{(x-1)^{6}} \)
d. \( 10 \displaystyle \frac{(x+1)^{4}}{(x-1)^{6}} \)
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