Stewart – Calculus ET 8e Chapter 13 Form A
1. Find the following limit.
2. Find the point of intersection of the tangent lines to the curve , at the
points where and .
3. Find an expression for .
4. The curves and intersects at the origin. Find their angle of
intersection correct to the nearest degree.
5. Find the derivative of the vector function.
6. Find
if and .
7. Find equations of the normal plane to at the point (2, 4, 8).
8. A particle moves with position function . Find the acceleration of the
particle.
9. A particle moves with position function .
Find the normal component of the acceleration vector.
10. Find the position vector of a particle that has the given acceleration and the given initial velocity
and position.
11. Sketch the curve of the vector function and indicate the orientation
of the curve.
12. Sketch the curve of the vector function , and indicate the orientation of the
curve.
Stewart – Calculus ET 8e Chapter 13 Form A
13. Sketch the curve of the vector function and indicate the orientation of the
curve.
14. Find the limit .
15. Sketch the curve of the vector function and indicate the orientation
of the curve.
16. Find the integral
17. Find the unit tangent and unit normal vectors and for the curve C defined by
Sketch the graph of C, and show and for
18. Find the velocity, acceleration, and speed of an object with position function
for Sketch the path of the object and its velocity and acceleration vectors.
19. Find the scalar tangential and normal components of acceleration of a particle with position vector
20. Find the velocity and position vectors of an object with acceleration
initial velocity and initial position
Stewart – Calculus ET 8e Chapter 13 Form A
Answer Key
Stewart – Calculus ET 8e Chapter 13 Form A
Stewart – Calculus ET 8e Chapter 13 Form A
Stewart – Calculus ET 8e Chapter 13 Form B
1. Find
for the function given.
2. Find an expression for .
3. Find equations of the normal plane to at the point (2, 4, 8).
4. Sketch the curve of the vector function , and indicate the orientation of the
curve.
5. Sketch the curve of the vector function and indicate the orientation of the
curve.
6. Find the limit .
7. Sketch the curve of the vector function and indicate the orientation
of the curve.
8. Sketch the curve of the vector function , and indicate the orientation of the
curve.
9. Find a vector function describing the curve of intersection of the cylinder and the
plane
10. Find
and for
11. Find
and for
12. Find the integral
13. Find the derivative
14. Given
a. Find and .
b. Sketch the curve defined by r and the vectors and on the same set of axes.
Stewart – Calculus ET 8e Chapter 13 Form B
15. Find the arc length function for the curve defined by for
Then use this result to find a parametrization of C in terms of s.
16. Find the velocity, acceleration, and speed of an object with position function
for Sketch the path of the object and its velocity and acceleration vectors.
17. A projectile is fired from a height of 350 ft with an initial speed of 200 ft/sec and an angle of
elevation of .
a. What are the scalar tangential and normal components of acceleration of the projectile?
b. What are the scalar tangential and normal components of acceleration of the projectile when the
projectile is at its maximum height?
18. Find the velocity, acceleration, and speed of an object with position vector .
19. Find the velocity, acceleration, and speed of an object with position vector .
20. Find the velocity and position vectors of an object with acceleration
initial velocity and initial position
Stewart – Calculus ET 8e Chapter 13 Form B
Answer Key
Stewart – Calculus ET 8e Chapter 13 Form B
Stewart – Calculus ET 8e Chapter 13 Form B
Stewart – Calculus ET 8e Chapter 13 Form C
Select the correct answer for each question.
____ 1. Find the limit.
a.
b.
c.
d.
e.
____ 2. Let .
Find the domain of .
a.
b.
c.
d.
e.
____ 3. Find the domain of the vector function .
a.
b.
c.
d.
Stewart – Calculus ET 8e Chapter 13 Form C
____ 4. Find a vector function that represents the curve of intersection of the two surfaces:
the top half of the ellipsoid and the parabolic cylinder .
a.
b.
c.
d.
e.
____ 5. Find a vector function that represents the curve of intersection of the two surfaces:
The circular cylinder and the parabolic cylinder .
a.
b.
c.
d.
e.
____ 6. If , find .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 13 Form C
____ 7. The helix intersects the curve at the point
. Find the angle of intersection.
a.
b.
c.
d.
e. None of these
____ 8. Find parametric equations for the tangent line to the curve with parametric equations
at the point with
a.
b.
c.
d.
Stewart – Calculus ET 8e Chapter 13 Form C
____ 9. Find the unit tangent vector for the curve given by .
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 13 Form C
____ 10. Reparametrize the curve with respect to arc length measured from the point where in the
direction of increasing .
a.
b.
c.
d.
e.
____ 11. Find the length of the curve
a.
214
b.
14
c.
614
d.
314
Stewart – Calculus ET 8e Chapter 13 Form C
____ 12. The torsion of a curve defined by is given by
Find the torsion of the curve defined by .
a.
21
41
b.
9
41
c.
20
41
d.
80
41
____ 13. Use Simpson’s Rule with n = 4 to estimate the length of the arc of the curve with equations
, from to . Round your answer to four decimal places.
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 13 Form C
____ 14. The curvature of the curve given by the vector function is
Use the formula to find the curvature of at the point .
a.
b.
c.
d.
e.
____ 15. Find the velocity of a particle with the given position function.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 13 Form C
____ 16. Find the speed of a particle with the given position function.
a.
b.
c.
d.
e.
____ 17. A mortar shell is fired with a muzzle speed of 250 ft/sec. Find the angle of elevation of the mortar
if the shell strikes a target located 1100 ft away. Round your answer to 2 decimal places.
a.
b.
c.
d.
____ 18. Find the scalar tangential and normal components of acceleration of a particle with position vector
a.
b.
c.
d.
____ 19. A projectile is fired with an initial speed of and angle of elevation . Find the range of
the projectile.
a. km
b. km
c. km
d. km
e. km
Stewart – Calculus ET 8e Chapter 13 Form C
____ 20. The following table gives coordinates of a particle moving through space along a smooth curve.
Find the average velocity over the time interval .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 13 Form C
Answer Key