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Deck 3: Differentiation
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Question
Suppose that 
is a function such that 
. What is 
?
is a function such that . What is ?
Question
Given a function 
, what does the quantity 
represent?
, what does the quantity represent? Question
If 
then
A)
B)
C)
D)
E)
thenA)
B)
C)
D)
E)
Question
An equation of the tangent line to the graph of 
at both 
and 
is 
. Which of the following statements must be true?
A) The function
is the line 
.
B)
C)
D) The function
is not differentiable at 
and at 
.
E) None of the above.
at both and is . Which of the following statements must be true?A) The function
is the line .B)
C)
D) The function
is not differentiable at and at .E) None of the above.
Question
True or False: The graph of the function 
has a vertical tangent line at 
.
has a vertical tangent line at . Question
What is the derivative of the function 
at the point where 
?
at the point where ? Question
Let 
be a function satisfying 
. Which of the following statements must be true?
A)
B)
for all 
C) The tangent line to the graph of
at every point has a positive slope.
D)
is not differentiable at 
E) None of the above.
be a function satisfying . Which of the following statements must be true?A)
B)
for all C) The tangent line to the graph of
at every point has a positive slope.D)
is not differentiable at E) None of the above.
Question
Let 
. Compute 
and find an equation of the tangent line to the graph of 
at 
. Compute and find an equation of the tangent line to the graph of at Question
If 
is a variable and 
is a constant, which of the following expressions equals 
?
A)
B)
C)
D)
E)
is a variable and is a constant, which of the following expressions equals ?A)
B)
C)
D)
E)
Question
Which of the following functions is continuous at 
but not differentiable there?
A)
B)
C)
D)
E) All of the above.
but not differentiable there?A)
B)
C)
D)
E) All of the above.
Question
If 
what is 
?
what is ? Question
What does the limit 
represent?
represent? Question
What is the derivative of 
?
A)
B)
C)
D)
E)
?A)
B)
C)
D)
E)
Question
We can use the difference quotient 
as an approximation to 
when 
and 
are known.
A) Find an approximation to
based on the values of 
and 
.
Given
such that: 
B) Use the approximation found in (A) to approximate
C) Use the approximation found in (A) to approximate
as an approximation to when and are known. A) Find an approximation to
based on the values of and .Given
such that: B) Use the approximation found in (A) to approximate
C) Use the approximation found in (A) to approximate
Question
Suppose 
on 
. If 
, 
, 
, 
, and 
, what is 
?
on . If , , , , and , what is ? Question
The derivative of the function 
at the point where 
equals
A)
B)
C)
D)
E) None of the above.
at the point where equalsA)
B)
C)
D)
E) None of the above.
Question
True or False: The derivative of 
is 
.
is . Question
Find the 
coordinates of all the points on the curve 
, besides 
, at which the slope of the tangent line equals 1.5.
coordinates of all the points on the curve , besides , at which the slope of the tangent line equals 1.5. Question
Find 
and 
if an equation of the tangent line to the graph of the function 
at 
is 
.
and if an equation of the tangent line to the graph of the function at is . Question
At which points is the function shown in the figure not differentiable? 
A)
and 
B)
and 
C) Only at
D) Only at
E) Only at
A)
and B)
and C) Only at
D) Only at
E) Only at
Question
At what rate is the volume of a hemisphere with radius 
changing when 
?
changing when ? Question
The figure shows the graph of the distance covered by a particle in meters, versus the time elapsed in minutes: 
True or False: The particle's velocity is maximal when 
.
True or False: The particle's velocity is maximal when . Question
The figure shows the graph of the distance covered by a particle in meters, versus the time elapsed in minutes: 
True or False: The particle's velocity is negative for more than one minute.
True or False: The particle's velocity is negative for more than one minute. Question
The dollar-cost of producing 
cars in a certain car factory is given by the formula 
.
A) Estimate the marginal cost at a production level of 3000 cars.
B) Compute the exact marginal cost.
cars in a certain car factory is given by the formula . A) Estimate the marginal cost at a production level of 3000 cars.
B) Compute the exact marginal cost.
Question
True or False: If 
for all 
then 
for all 
.
for all then for all . Question
What is the equation of the tangent line to the graph of 
at 
, if 
?
A)
B)
C)
D)
E) None of the above.
at , if ?A)
B)
C)
D)
E) None of the above.
Question
The height (in meters) at time 
(in hours) of a balloon is given by the formula 
.
A) What is the balloon's height when it starts moving?
B) What is the balloon's velocity after 30 minutes?
C) Find the maximum height of the balloon.
(in hours) of a balloon is given by the formula . A) What is the balloon's height when it starts moving?
B) What is the balloon's velocity after 30 minutes?
C) Find the maximum height of the balloon.
Question
Let 
where 
is a constant and 
is a variable. Which of the following equals 
?
A)
B)
C)
D)
E)
where is a constant and is a variable. Which of the following equals ?A)
B)
C)
D)
E)
Question
What is the speed of a stone dropped from a building of height 320 m when it hits the ground? (use 
)
) Question
Find the equation of the tangent line to the graph of the function 
at 
.
at . Question
Let 
and 
be differentiable functions at 
and 
. If 
which of the following statements must be true?
A)
B)
C)
D)
E)
and be differentiable functions at and . If which of the following statements must be true?A)
B)
C)
D)
E)
Question
Given the function 
, which of the following statements is correct?
A) The slopes of the tangent lines to the graph of this function get steeper as
increases.
B) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to 1.
C) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to 
.
D) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to 
.
E) The slopes of the tangent lines to the graph of this function approach 0 as
approaches 0.
, which of the following statements is correct?A) The slopes of the tangent lines to the graph of this function get steeper as
increases.B) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to 1.C) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to .D) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to .E) The slopes of the tangent lines to the graph of this function approach 0 as
approaches 0. Question
If 
and 
is a natural number then 
:
A) For all values of
B) For
C) For
D) For
E)
for all values of 
and is a natural number then :A) For all values of
B) For
C) For
D) For
E)
for all values of Question
Let 
be the dollar cost of 
products. The marginal cost of producing 
products is
A)
B)
C)
D)
E)
be the dollar cost of products. The marginal cost of producing products isA)
B)
C)
D)
E)
Question
By Ohm's Law, the voltage 
current 
and the resistance 
are related by the equation 
, where the units of the voltage, current, and resistance are volts, amperes and ohms, respectively. Assume the resistance is constant and equals 
. What is the average rate of change of 
with respect to 
for the interval 
to 
.
A)
amperes/volt
B)
amperes/volt
C)
amperes/volt
D)
amperes/volt
E)
amperes/volt
current and the resistance are related by the equation , where the units of the voltage, current, and resistance are volts, amperes and ohms, respectively. Assume the resistance is constant and equals . What is the average rate of change of with respect to for the interval to .A)
amperes/voltB)
amperes/voltC)
amperes/voltD)
amperes/voltE)
amperes/volt Question
True or False: If both 
and 
are differentiable at 
but 
then 
is not differentiable at 
.
and are differentiable at but then is not differentiable at . Question
The function 
satisfies: 
. Find 
where 
.
satisfies: . Find where . Question
Let 
. Find 
.
A)
B)
C) Not defined because we cannot use the Quotient Rule near
D) Not defined because
does not exist
E) Not defined because
is not continuous at 
. Find .A)
B)
C) Not defined because we cannot use the Quotient Rule near
D) Not defined because
does not existE) Not defined because
is not continuous at Question
Given 
, find 
.
, find . Question
Calculate the derivatives of the following functions:
A)
B)
C)
A)
B)
C)
Question
If 
then 
?
then ? Question
Find the 
coordinates of all points on the curve 
, 
, where the tangent line is horizontal.
coordinates of all points on the curve , , where the tangent line is horizontal. Question
The function 
satisfies the following initial value problem:
A)
B)
C)
D)
E) None of the above.
satisfies the following initial value problem:A)
B)
C)
D)
E) None of the above.
Question
Which of the following is a correct formula for 
?
A)
B)
C)
D)
E)
?A)
B)
C)
D)
E)
Question
and 
are two differentiable functions satisfying 
and 
. Which of the following formulas represents 
?A) 0
B)
C)
D)
E)
Question
The graph of 
is given below. Sketch the graphs of 
and 
. 
is given below. Sketch the graphs of and . Question
? Question
What is the equation of the tangent line to the function 
at 
?
at ? Question
? Question
If 
is the tangent line to the graph of the function 
at 
then what is the slope of the tangent line to the graph of the function 
at 
?
is the tangent line to the graph of the function at then what is the slope of the tangent line to the graph of the function at ? Question
Compute the derivatives of the following functions:
A)
B)
C)
D)
A)
B)
C)
D)
Question
What is an equation of the tangent line to the graph of 
at 
?
at ? Question
Find the 
coordinates of all points on the curve 
, 
, where the tangent line is parallel to the line 
.
coordinates of all points on the curve , , where the tangent line is parallel to the line . Question
What is the second derivative of the volume of a ball with respect to its radius 
?
A)
B)
C)
D) 0
E)
?A)
B)
C)
D) 0
E)
Question
Which of the following statements about the function 
must be incorrect?
A) It is differentiable for positive values of
.
B) Its derivative has an infinite number of zeros.
C) If
then 
satisfies 
.
D) Its derivative is a positive function for positive values of
.
E) For
, 
.
must be incorrect?A) It is differentiable for positive values of
.B) Its derivative has an infinite number of zeros.
C) If
then satisfies .D) Its derivative is a positive function for positive values of
.E) For
, . Question
For what values of the natural number 
is the formula 
valid?
A) None
B) for all values of
C) for
even
D) for
odd
E) for k such that a remainder of 3 is obtained upon division of
by 4
is the formula valid?A) None
B) for all values of
C) for
evenD) for
oddE) for k such that a remainder of 3 is obtained upon division of
by 4 Question
Which of the following is a correct formula for 
?
A)
B)
C)
D)
E) All of the above are identical and correct.
?A)
B)
C)
D)
E) All of the above are identical and correct.
Question
Let 
. What is the value of 
?
. What is the value of ? Question
The graphs of 
and 
are shown below. Determine which graph corresponds to 
and 
. 
and are shown below. Determine which graph corresponds to and . Question
Find 
Question
A right circular cylinder of height 10 meters and radius 4 meters is initially filled with water. The water leaks out of a hole in the bottom at a rate of 
cubic meters per minute where 
is the height of the water remaining in the tank. At what rate is the height of the water decreasing when 
meters?
cubic meters per minute where is the height of the water remaining in the tank. At what rate is the height of the water decreasing when meters? Question
Given the equation 
, find the correct statement.
A)
is a function of 
.
B) The graph of the equation is symmetric with respect to the y axis.
C)
is a point on the graph of the equation.
D) The tangent line at
has slope 
.
E) The graph of the equation crosses the x-axis at three different points.
, find the correct statement.A)
is a function of .B) The graph of the equation is symmetric with respect to the y axis.
C)
is a point on the graph of the equation.D) The tangent line at
has slope .E) The graph of the equation crosses the x-axis at three different points.
Question
Which of the following is incorrect with regard to the graph of 
( 
)?
A) The graph passes through the point
.
B) If
, then the graph is 
.
C) The tangent line to the graph at the point
is horizontal.
D) If
is even then the graph passes through the point 
.
E) None of the above.
( )?A) The graph passes through the point
.B) If
, then the graph is .C) The tangent line to the graph at the point
is horizontal.D) If
is even then the graph passes through the point .E) None of the above.
Question
True or False: If 
satisfies 
then any differentiable function 
satisfies 
. (Assume that 
is defined for all real numbers)
satisfies then any differentiable function satisfies . (Assume that is defined for all real numbers) Question
Which of the following is incorrect?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Let 
and 
. Which of the following statements is correct?
A)
and 
are not differentiable.
B) In
, 
has more zeros than 
.
C) In
, 
has less zeros than 
.
D) In
, 
has the same number of zeros as 
.
E) There exists a value of
such that 
.
and . Which of the following statements is correct?A)
and are not differentiable.B) In
, has more zeros than .C) In
, has less zeros than .D) In
, has the same number of zeros as .E) There exists a value of
such that . Question
A 2 meter ladder is leaning against a wall. At 
, the height of the projection of the ladder on the wall is 1 meter, and it starts sliding away from the wall at a rate
of
m/sec. Find the velocity of the top of the ladder at 
sec.
, the height of the projection of the ladder on the wall is 1 meter, and it starts sliding away from the wall at a rateof
m/sec. Find the velocity of the top of the ladder at sec. Question
True or False: If 
satisfies 
then any differentiable function 
satisfies 
. (Assume that 
is defined for all real numbers)
satisfies then any differentiable function satisfies . (Assume that is defined for all real numbers) Question
A right triangle initially has side lengths of 3, 4, and 5 cm. Suppose the small side is increased at a rate of 1 cm/sec, while the length of the height is kept constant. How fast is the smallest angle of the right triangle changing when it reaches the value of 
rad ?
A)
rad/sec
B)
rad/sec
C)
rad/sec
D) We cannot answer this question because the function representing the smallest angle is not differentiable.
E) We cannot answer this question because it depends on the time which is unknown.
rad ?A)
rad/secB)
rad/secC)
rad/secD) We cannot answer this question because the function representing the smallest angle is not differentiable.
E) We cannot answer this question because it depends on the time which is unknown.
Question
Find 
at the point 
if 
at the point if Question
The pressure ( 
) and volume ( 
) of an expanding gas are related by the formula 
.What is the value of 
if 
and 
.
A) The value cannot be found because we are not given the time
.
B) The value cannot be found because we are not given the pressure
.
C) Both A and B are correct.
D) Approximately
E) Approximately
) and volume ( ) of an expanding gas are related by the formula .What is the value of if and .A) The value cannot be found because we are not given the time
.B) The value cannot be found because we are not given the pressure
.C) Both A and B are correct.
D) Approximately
E) Approximately
Question
The sides of a rectangle are increasing. One side is increasing at a rate of 3 m/sec and the adjacent side is increasing at a rate of 4 m/sec. At what rate is the area of the rectangle increasing?
A) 12
B) 7
C) 25
D) 5
E) The solution depends on the values of the sides at the time
.
A) 12
B) 7
C) 25
D) 5
E) The solution depends on the values of the sides at the time
. Question
Question
Let V be the volume of a cylinder with height 
and radius 
. Suppose 
increases with time and 
decreases with time. If the volume does not change what is the relationship between 
and 
?
and radius . Suppose increases with time and decreases with time. If the volume does not change what is the relationship between and ? Question
Let 
. Find 
.
. Find . Question
Find the points on the graph of 
where the tangent line is horizontal.
where the tangent line is horizontal. Question
True or False: There exists a natural number 
such that the tangent line to the
graph of the equation
has slope 1.1 at the point 
.
such that the tangent line to thegraph of the equation
has slope 1.1 at the point . Question
A point 
is moving along the curve 
. When 
reaches the point 
is increasing at a rate of 
unit/sec. At what rate is 
increasing?
A)
B)
C)
D)
E) We cannot solve this question because we do not know the value of
.
is moving along the curve . When reaches the point is increasing at a rate of unit/sec. At what rate is increasing?A)
B)
C)
D)
E) We cannot solve this question because we do not know the value of
. Question
What is the equation of the tangent line to 
at the point 
?
at the point ? Question
Find a point besides the origin where the graph of 
has a vertical tangent.
has a vertical tangent.
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Full screen (f)
Deck 3: Differentiation
1
Suppose that is a function such that . What is ?
is a function such that . What is ?
2
Given a function , what does the quantity represent?
, what does the quantity represent?The slope of the secant line to the graph of 
containing the points on the graph of 
where 
and 
.
containing the points on the graph of where and . 3
If then
A)
B)
C)
D)
E)
thenA)
B)
C)
D)
E)
4
An equation of the tangent line to the graph of at both and is . Which of the following statements must be true?
A) The function is the line .
B)
C)
D) The function is not differentiable at and at .
E) None of the above.
at both and is . Which of the following statements must be true?A) The function
is the line .B)
C)
D) The function
is not differentiable at and at .E) None of the above.
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5
True or False: The graph of the function has a vertical tangent line at .
has a vertical tangent line at . Unlock Deck
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6
What is the derivative of the function at the point where ?
at the point where ? Unlock Deck
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7
Let be a function satisfying . Which of the following statements must be true?
A)
B) for all
C) The tangent line to the graph of at every point has a positive slope.
D) is not differentiable at
E) None of the above.
be a function satisfying . Which of the following statements must be true?A)
B)
for all C) The tangent line to the graph of
at every point has a positive slope.D)
is not differentiable at E) None of the above.
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8
Let . Compute and find an equation of the tangent line to the graph of at
. Compute and find an equation of the tangent line to the graph of at Unlock Deck
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9
If is a variable and is a constant, which of the following expressions equals ?
A)
B)
C)
D)
E)
is a variable and is a constant, which of the following expressions equals ?A)
B)
C)
D)
E)
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10
Which of the following functions is continuous at but not differentiable there?
A)
B)
C)
D)
E) All of the above.
but not differentiable there?A)
B)
C)
D)
E) All of the above.
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11
If what is ?
what is ? Unlock Deck
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12
What does the limit represent?
represent? Unlock Deck
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13
What is the derivative of ?
A)
B)
C)
D)
E)
?A)
B)
C)
D)
E)
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14
We can use the difference quotient as an approximation to when and are known.
A) Find an approximation to based on the values of and .
Given such that:
B) Use the approximation found in (A) to approximate
C) Use the approximation found in (A) to approximate
as an approximation to when and are known. A) Find an approximation to
based on the values of and .Given
such that: B) Use the approximation found in (A) to approximate
C) Use the approximation found in (A) to approximate
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15
Suppose on . If , , , , and , what is ?
on . If , , , , and , what is ? Unlock Deck
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16
The derivative of the function at the point where equals
A)
B)
C)
D)
E) None of the above.
at the point where equalsA)
B)
C)
D)
E) None of the above.
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17
True or False: The derivative of is .
is . Unlock Deck
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18
Find the coordinates of all the points on the curve , besides , at which the slope of the tangent line equals 1.5.
coordinates of all the points on the curve , besides , at which the slope of the tangent line equals 1.5. Unlock Deck
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19
Find and if an equation of the tangent line to the graph of the function at is .
and if an equation of the tangent line to the graph of the function at is . Unlock Deck
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20
At which points is the function shown in the figure not differentiable?
A) and
B) and
C) Only at
D) Only at
E) Only at
A)
and B)
and C) Only at
D) Only at
E) Only at
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21
At what rate is the volume of a hemisphere with radius changing when ?
changing when ? Unlock Deck
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22
The figure shows the graph of the distance covered by a particle in meters, versus the time elapsed in minutes: True or False: The particle's velocity is maximal when .
True or False: The particle's velocity is maximal when . Unlock Deck
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23
The figure shows the graph of the distance covered by a particle in meters, versus the time elapsed in minutes: True or False: The particle's velocity is negative for more than one minute.
True or False: The particle's velocity is negative for more than one minute. Unlock Deck
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24
The dollar-cost of producing cars in a certain car factory is given by the formula .
A) Estimate the marginal cost at a production level of 3000 cars.
B) Compute the exact marginal cost.
cars in a certain car factory is given by the formula . A) Estimate the marginal cost at a production level of 3000 cars.
B) Compute the exact marginal cost.
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25
True or False: If for all then for all .
for all then for all . Unlock Deck
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26
What is the equation of the tangent line to the graph of at , if ?
A)
B)
C)
D)
E) None of the above.
at , if ?A)
B)
C)
D)
E) None of the above.
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27
The height (in meters) at time (in hours) of a balloon is given by the formula .
A) What is the balloon's height when it starts moving?
B) What is the balloon's velocity after 30 minutes?
C) Find the maximum height of the balloon.
(in hours) of a balloon is given by the formula . A) What is the balloon's height when it starts moving?
B) What is the balloon's velocity after 30 minutes?
C) Find the maximum height of the balloon.
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28
Let where is a constant and is a variable. Which of the following equals ?
A)
B)
C)
D)
E)
where is a constant and is a variable. Which of the following equals ?A)
B)
C)
D)
E)
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29
What is the speed of a stone dropped from a building of height 320 m when it hits the ground? (use )
) Unlock Deck
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30
Find the equation of the tangent line to the graph of the function at .
at . Unlock Deck
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31
Let and be differentiable functions at and . If which of the following statements must be true?
A)
B)
C)
D)
E)
and be differentiable functions at and . If which of the following statements must be true?A)
B)
C)
D)
E)
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32
Given the function , which of the following statements is correct?
A) The slopes of the tangent lines to the graph of this function get steeper as increases.
B) The rate of change of the slopes of the tangent lines to the graph of this function at is equal to 1.
C) The rate of change of the slopes of the tangent lines to the graph of this function at is equal to .
D) The rate of change of the slopes of the tangent lines to the graph of this function at is equal to .
E) The slopes of the tangent lines to the graph of this function approach 0 as approaches 0.
, which of the following statements is correct?A) The slopes of the tangent lines to the graph of this function get steeper as
increases.B) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to 1.C) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to .D) The rate of change of the slopes of the tangent lines to the graph of this function at
is equal to .E) The slopes of the tangent lines to the graph of this function approach 0 as
approaches 0. Unlock Deck
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33
If and is a natural number then :
A) For all values of
B) For
C) For
D) For
E) for all values of
and is a natural number then :A) For all values of
B) For
C) For
D) For
E)
for all values of Unlock Deck
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34
Let be the dollar cost of products. The marginal cost of producing products is
A)
B)
C)
D)
E)
be the dollar cost of products. The marginal cost of producing products isA)
B)
C)
D)
E)
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35
By Ohm's Law, the voltage current and the resistance are related by the equation , where the units of the voltage, current, and resistance are volts, amperes and ohms, respectively. Assume the resistance is constant and equals . What is the average rate of change of with respect to for the interval to .
A) amperes/volt
B) amperes/volt
C) amperes/volt
D) amperes/volt
E) amperes/volt
current and the resistance are related by the equation , where the units of the voltage, current, and resistance are volts, amperes and ohms, respectively. Assume the resistance is constant and equals . What is the average rate of change of with respect to for the interval to .A)
amperes/voltB)
amperes/voltC)
amperes/voltD)
amperes/voltE)
amperes/volt Unlock Deck
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36
True or False: If both and are differentiable at but then is not differentiable at .
and are differentiable at but then is not differentiable at . Unlock Deck
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37
The function satisfies: . Find where .
satisfies: . Find where . Unlock Deck
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38
Let . Find .
A)
B)
C) Not defined because we cannot use the Quotient Rule near
D) Not defined because does not exist
E) Not defined because is not continuous at
. Find .A)
B)
C) Not defined because we cannot use the Quotient Rule near
D) Not defined because
does not existE) Not defined because
is not continuous at Unlock Deck
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39
Given , find .
, find . Unlock Deck
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40
Calculate the derivatives of the following functions:
A)
B)
C)
A)
B)
C)
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41
If then ?
then ? Unlock Deck
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42
Find the coordinates of all points on the curve , , where the tangent line is horizontal.
coordinates of all points on the curve , , where the tangent line is horizontal. Unlock Deck
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43
The function satisfies the following initial value problem:
A)
B)
C)
D)
E) None of the above.
satisfies the following initial value problem:A)
B)
C)
D)
E) None of the above.
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44
Which of the following is a correct formula for ?
A)
B)
C)
D)
E)
?A)
B)
C)
D)
E)
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45
and are two differentiable functions satisfying and . Which of the following formulas represents ?A) 0
B)
C)
D)
E)
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46
The graph of is given below. Sketch the graphs of and .
is given below. Sketch the graphs of and . Unlock Deck
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47
? Unlock Deck
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48
What is the equation of the tangent line to the function at ?
at ? Unlock Deck
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49
? Unlock Deck
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50
If is the tangent line to the graph of the function at then what is the slope of the tangent line to the graph of the function at ?
is the tangent line to the graph of the function at then what is the slope of the tangent line to the graph of the function at ? Unlock Deck
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51
Compute the derivatives of the following functions:
A)
B)
C)
D)
A)
B)
C)
D)
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52
What is an equation of the tangent line to the graph of at ?
at ? Unlock Deck
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53
Find the coordinates of all points on the curve , , where the tangent line is parallel to the line .
coordinates of all points on the curve , , where the tangent line is parallel to the line . Unlock Deck
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54
What is the second derivative of the volume of a ball with respect to its radius ?
A)
B)
C)
D) 0
E)
?A)
B)
C)
D) 0
E)
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55
Which of the following statements about the function must be incorrect?
A) It is differentiable for positive values of .
B) Its derivative has an infinite number of zeros.
C) If then satisfies .
D) Its derivative is a positive function for positive values of .
E) For , .
must be incorrect?A) It is differentiable for positive values of
.B) Its derivative has an infinite number of zeros.
C) If
then satisfies .D) Its derivative is a positive function for positive values of
.E) For
, . Unlock Deck
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56
For what values of the natural number is the formula valid?
A) None
B) for all values of
C) for even
D) for odd
E) for k such that a remainder of 3 is obtained upon division of by 4
is the formula valid?A) None
B) for all values of
C) for
evenD) for
oddE) for k such that a remainder of 3 is obtained upon division of
by 4 Unlock Deck
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57
Which of the following is a correct formula for ?
A)
B)
C)
D)
E) All of the above are identical and correct.
?A)
B)
C)
D)
E) All of the above are identical and correct.
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58
Let . What is the value of ?
. What is the value of ? Unlock Deck
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59
The graphs of and are shown below. Determine which graph corresponds to and .
and are shown below. Determine which graph corresponds to and . Unlock Deck
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60
Find
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61
A right circular cylinder of height 10 meters and radius 4 meters is initially filled with water. The water leaks out of a hole in the bottom at a rate of cubic meters per minute where is the height of the water remaining in the tank. At what rate is the height of the water decreasing when meters?
cubic meters per minute where is the height of the water remaining in the tank. At what rate is the height of the water decreasing when meters? Unlock Deck
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62
Given the equation , find the correct statement.
A) is a function of .
B) The graph of the equation is symmetric with respect to the y axis.
C) is a point on the graph of the equation.
D) The tangent line at has slope .
E) The graph of the equation crosses the x-axis at three different points.
, find the correct statement.A)
is a function of .B) The graph of the equation is symmetric with respect to the y axis.
C)
is a point on the graph of the equation.D) The tangent line at
has slope .E) The graph of the equation crosses the x-axis at three different points.
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63
Which of the following is incorrect with regard to the graph of ( )?
A) The graph passes through the point .
B) If , then the graph is .
C) The tangent line to the graph at the point is horizontal.
D) If is even then the graph passes through the point .
E) None of the above.
( )?A) The graph passes through the point
.B) If
, then the graph is .C) The tangent line to the graph at the point
is horizontal.D) If
is even then the graph passes through the point .E) None of the above.
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64
True or False: If satisfies then any differentiable function satisfies . (Assume that is defined for all real numbers)
satisfies then any differentiable function satisfies . (Assume that is defined for all real numbers) Unlock Deck
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65
Which of the following is incorrect?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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66
Let and . Which of the following statements is correct?
A) and are not differentiable.
B) In , has more zeros than .
C) In , has less zeros than .
D) In , has the same number of zeros as .
E) There exists a value of such that .
and . Which of the following statements is correct?A)
and are not differentiable.B) In
, has more zeros than .C) In
, has less zeros than .D) In
, has the same number of zeros as .E) There exists a value of
such that . Unlock Deck
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67
A 2 meter ladder is leaning against a wall. At , the height of the projection of the ladder on the wall is 1 meter, and it starts sliding away from the wall at a rate
of m/sec. Find the velocity of the top of the ladder at sec.
, the height of the projection of the ladder on the wall is 1 meter, and it starts sliding away from the wall at a rateof
m/sec. Find the velocity of the top of the ladder at sec. Unlock Deck
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68
True or False: If satisfies then any differentiable function satisfies . (Assume that is defined for all real numbers)
satisfies then any differentiable function satisfies . (Assume that is defined for all real numbers) Unlock Deck
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69
A right triangle initially has side lengths of 3, 4, and 5 cm. Suppose the small side is increased at a rate of 1 cm/sec, while the length of the height is kept constant. How fast is the smallest angle of the right triangle changing when it reaches the value of rad ?
A) rad/sec
B) rad/sec
C) rad/sec
D) We cannot answer this question because the function representing the smallest angle is not differentiable.
E) We cannot answer this question because it depends on the time which is unknown.
rad ?A)
rad/secB)
rad/secC)
rad/secD) We cannot answer this question because the function representing the smallest angle is not differentiable.
E) We cannot answer this question because it depends on the time which is unknown.
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70
Find at the point if
at the point if Unlock Deck
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71
The pressure ( ) and volume ( ) of an expanding gas are related by the formula .What is the value of if and .
A) The value cannot be found because we are not given the time .
B) The value cannot be found because we are not given the pressure .
C) Both A and B are correct.
D) Approximately
E) Approximately
) and volume ( ) of an expanding gas are related by the formula .What is the value of if and .A) The value cannot be found because we are not given the time
.B) The value cannot be found because we are not given the pressure
.C) Both A and B are correct.
D) Approximately
E) Approximately
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72
The sides of a rectangle are increasing. One side is increasing at a rate of 3 m/sec and the adjacent side is increasing at a rate of 4 m/sec. At what rate is the area of the rectangle increasing?
A) 12
B) 7
C) 25
D) 5
E) The solution depends on the values of the sides at the time .
A) 12
B) 7
C) 25
D) 5
E) The solution depends on the values of the sides at the time
. Unlock Deck
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73
True or False: If the radius of a circle is increasing with time then the area of the circle is increasing at the same rate.
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74
Let V be the volume of a cylinder with height and radius . Suppose increases with time and decreases with time. If the volume does not change what is the relationship between and ?
and radius . Suppose increases with time and decreases with time. If the volume does not change what is the relationship between and ? Unlock Deck
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75
Let . Find .
. Find . Unlock Deck
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76
Find the points on the graph of where the tangent line is horizontal.
where the tangent line is horizontal. Unlock Deck
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77
True or False: There exists a natural number such that the tangent line to the
graph of the equation has slope 1.1 at the point .
such that the tangent line to thegraph of the equation
has slope 1.1 at the point . Unlock Deck
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78
A point is moving along the curve . When reaches the point is increasing at a rate of unit/sec. At what rate is increasing?
A)
B)
C)
D)
E) We cannot solve this question because we do not know the value of .
is moving along the curve . When reaches the point is increasing at a rate of unit/sec. At what rate is increasing?A)
B)
C)
D)
E) We cannot solve this question because we do not know the value of
. Unlock Deck
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79
What is the equation of the tangent line to at the point ?
at the point ? Unlock Deck
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80
Find a point besides the origin where the graph of has a vertical tangent.
has a vertical tangent. Unlock Deck
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