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3.5 Implicit Differentiation
169
____ 19. Find an equation of the tangent line to the graph of the function
places.
a.
b.
c.
d.
e.
at the point .The coefficients below are given to two decimal
____ 20. Use implicit differentiation to find an equation of the tangent line to the ellipse
at .
a.
b.
c.
d.
e.
____ 21. Find in terms of x and y given that . Use the original equation to simplify
your answer.
a.
b.
c.
d.
e.
170
Chapter 3: Differentiation
____
22.
Find
in terms of x and y given that
a.
b.
c.
d.
e.
____
23.
Find
in terms of x and y.
a.
b.
c.
d.
e.
3.5 Implicit Differentiation
171
____ 24. Find in terms of x and y.
a.
b.
c.
d.
e.
____ 25. Find the points at which the graph of the equation has a vertical or horizontal tangent
line.
There is a vertical tangent at but no horizontal tangents.
There is a horizontal tangent at and a vertical tangent at .
There is a horizontal tangent at but no vertical tangents.
There is a horizontal tangent at and a vertical tangent at .
There are no horizontal or vertical tangent lines.
172 Chapter 3: Differentiation
____ 26. Use logarithmic differentiation to find the derivative of .
a.
b.
c.
d.
e.
____ 27. Use logarithmic differentiation to find .
a.
b.
c.
d.
e.
3.5 Implicit Differentiation
173
____ 28. Use logarithmic differentiation to find .
a.
b.
c.
d.
e.
____ 29. Differentiate with respect to t (x and y are functions of t).
a.
b.
c.
d.
e.
____ 30. Differentiate with respect to t (x and y are functions of t).
a.
b.
c.
d.
e.
Chapter 3: Differentiation
Implicit Differentiation
Answer Section
3.5 Implicit Differentiation
175
176 Chapter 3: Differentiation
3.6 Derivatives of Inverse Functions
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find an equation of the tangent line to the graph of at the point
.
a.
b.
c.
d.
e.
____ 2. Find at the point for the equation .
a.
b.
c.
d.
e.
____ 3. Find at the point for the equation .
a.
b.
c.
d.
e.
3.6 Derivatives of Inverse Functions
177
____
4.
Find the derivative of the function
.
a.
b.
c.
d.
e.
____
5.
Find the derivative of the function
.
a.
b.
c.
d.
e.
178 Chapter 3: Differentiation
____ 6. Find the derivative of the function .
a.
b.
c.
d.
e.
____ 7. Find the derivative of the function .
a.
b.
c.
d.
e.
3.6 Derivatives of Inverse Functions
179
____ 8.
Find the derivative of the function
.
a.
b.
c.
d.
e.
____ 9. Find the derivative of the function .
a.
b.
c.
d.
e.
180 Chapter 3: Differentiation
____ 10. Find the slope-intercept form of the equation of the line tangent to the graph of
.
a.
b.
c.
d.
e.
____ 11.
An airplane flies at an altitude of 14 miles toward a point directly over an observer.
Consider
and x as shown in the following figure. Write
as a function of x.
a.
b.
c.
d.
e.
3.6 Derivatives of Inverse Functions
181
____ 12.
An airplane flies at an altitude of 6 miles towards a point directly over an observer.
Consider
and x as shown in the following figure. The speed of the plane is 400 miles per hour.
Find when . Round your answer to three decimal places.
–17.647 rad/hr
2.941 rad/hr
17.647 rad/hr
0.044 rad/hr
–8.824 rad/hr
____ 13. In a free- fall experiment, an object is dropped from a height of 256 feet. A camera on the
ground 500 feet from the point of impact records the fall of the object as shown in the figure.
Assuming the object is released at time t = 0. At what time will the object reach the ground level?
3 seconds
6 seconds
7 seconds
5 seconds
4 seconds
182
Chapter 3: Differentiation
____
14.
In a free-fall experiment, an object is dropped from a height of 144 feet. A camera on
the ground 500 feet from the point of impact records the fall of the object as shown in the figure.
Assuming the object is released at time t = 0. Find the rate of change of the angle of elevation of the
camera when . Round your answer to four decimal places.
–0.6006 rad/sec
–0.0581 rad/sec
0.0601 rad/sec
0.0581 rad/sec
–0.0601 rad/sec
____ 15. A petrol car is parked 40 feet from a long warehouse (see figure). The revolving light on top
of the car turns at a rate of 30 revolutions per minute. Write as a function of x.
3.6 Derivatives of Inverse Functions
183
a.
b.
c.
d.
e.
____ 16. A petrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top
of the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the
wall when the beam makes an angle of with the perpendicular from the light to the wall.
a.
b.
c.
d.
e.
184 Chapter 3: Differentiation
3.6 Derivatives of Inverse Functions
Answer Section
