Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Chapter 13.1, Problem 45E
To determine
To find:
A
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A graph of the function f is given below:
Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1
Of is defined at a.
If is not defined at x = a.
Of is continuous at x = a.
If is discontinuous at x = a.
Of is smooth at x = a.
Of is not smooth at = a.
If has a horizontal tangent line at = a.
f has a vertical tangent line at x = a.
Of has a oblique/slanted tangent line at x = a.
If has no tangent line at x = a.
f(a + h) - f(a)
lim
is finite.
h→0
h
f(a + h) - f(a)
lim
h->0+
and lim
h
h->0-
f(a + h) - f(a)
h
are infinite.
lim
does not exist.
h→0
f(a+h) - f(a)
h
f'(a) is defined.
f'(a) is undefined.
If is differentiable at x = a.
If is not differentiable at x = a.
The graph below is the function f(z)
4
3
-2
-1
-1
1
2
3
-3
Consider the function f whose graph is given above.
(A) Find the following. If a function value is undefined, enter "undefined". If a limit does not exist, enter
"DNE". If a limit can be represented by -∞o or ∞o, then do so.
lim f(z)
+3
lim f(z)
1-1
lim f(z)
f(1)
= 2
=
-4
= undefined
lim f(z) 1
2-1
lim f(z):
2-1+
lim f(x)
2+1
-00
= -2
= DNE
f(-1) = -2
lim f(z) = -2
1-4
lim f(z)
2-4°
00
f'(0)
f'(2)
=
=
(B) List the value(s) of x for which f(x) is discontinuous. Then list the value(s) of x for which f(x) is left-
continuous or right-continuous. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5). If
there are none, enter "none".
Discontinuous at z =
Left-continuous at x =
Invalid use of a comma.syntax incomplete.
Right-continuous at z =
Invalid use of a comma.syntax incomplete.
(C) List the value(s) of x for which f(x) is non-differentiable. Enter your answer as a comma-separated list,
if needed (eg. -2, 3, 5).…
A graph of the function f is given below:
Study the graph of f at the value given below. Select each of the following that applies for the value
a = -4.
f is defined at = a.
f is not defined at 2 = a.
If is continuous at x = a.
Of is discontinuous at x = a.
Of is smooth at x = a.
f is not smooth at x = a.
If has a horizontal tangent line at x = a.
f has a vertical tangent line at x = a.
Of has a oblique/slanted tangent line at x = a.
Of has no tangent line at x = a.
f(a + h) − f(a)
h
lim
is finite.
h→0
f(a + h) - f(a)
lim
is infinite.
h→0
h
f(a + h) - f(a)
lim
does not exist.
h→0
h
f'(a) is defined.
f'(a) is undefined.
If is differentiable at x = a.
If is not differentiable at x = a.
Chapter 13 Solutions
Calculus (MindTap Course List)
Ch. 13.1 - Prob. 1ECh. 13.1 - Prob. 2ECh. 13.1 - Prob. 3ECh. 13.1 - Find the limit. limt1(t2tt1i+t+8j+sintlntk)Ch. 13.1 - Prob. 5ECh. 13.1 - Prob. 6ECh. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Prob. 8ECh. 13.1 - Prob. 9ECh. 13.1 - Prob. 10E
Ch. 13.1 - Prob. 11ECh. 13.1 - Prob. 12ECh. 13.1 - Prob. 13ECh. 13.1 - Prob. 14ECh. 13.1 - Prob. 15ECh. 13.1 - Prob. 16ECh. 13.1 - Prob. 17ECh. 13.1 - Prob. 18ECh. 13.1 - Prob. 19ECh. 13.1 - Prob. 20ECh. 13.1 - Prob. 21ECh. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Prob. 25ECh. 13.1 - Prob. 26ECh. 13.1 - Prob. 27ECh. 13.1 - Prob. 28ECh. 13.1 - Prob. 29ECh. 13.1 - Prob. 30ECh. 13.1 - Prob. 31ECh. 13.1 - Prob. 32ECh. 13.1 - Prob. 33ECh. 13.1 - Prob. 34ECh. 13.1 - Prob. 35ECh. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Prob. 40ECh. 13.1 - Show that the curve with parametric equations...Ch. 13.1 - Prob. 42ECh. 13.1 - Prob. 43ECh. 13.1 - Prob. 44ECh. 13.1 - Prob. 45ECh. 13.1 - Prob. 46ECh. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Prob. 50ECh. 13.1 - a Graph the curve with parametric equations...Ch. 13.1 - Prob. 52ECh. 13.1 - Prob. 53ECh. 13.1 - Prob. 54ECh. 13.2 - Prob. 1ECh. 13.2 - Prob. 2ECh. 13.2 - Prob. 3ECh. 13.2 - Prob. 4ECh. 13.2 - Prob. 5ECh. 13.2 - Prob. 6ECh. 13.2 - Prob. 7ECh. 13.2 - Prob. 8ECh. 13.2 - Prob. 9ECh. 13.2 - Prob. 10ECh. 13.2 - Prob. 11ECh. 13.2 - Prob. 12ECh. 13.2 - Prob. 13ECh. 13.2 - Prob. 14ECh. 13.2 - Prob. 15ECh. 13.2 - Prob. 16ECh. 13.2 - Prob. 17ECh. 13.2 - Prob. 18ECh. 13.2 - Prob. 19ECh. 13.2 - Prob. 20ECh. 13.2 - Prob. 21ECh. 13.2 - Prob. 22ECh. 13.2 - Prob. 23ECh. 13.2 - Prob. 24ECh. 13.2 - Prob. 25ECh. 13.2 - Prob. 26ECh. 13.2 - Prob. 27ECh. 13.2 - Prob. 28ECh. 13.2 - Prob. 29ECh. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Prob. 31ECh. 13.2 - Prob. 32ECh. 13.2 - Prob. 33ECh. 13.2 - Prob. 34ECh. 13.2 - Evaluate the integral. 02(tit3j+3t5k)dtCh. 13.2 - Prob. 36ECh. 13.2 - Prob. 37ECh. 13.2 - Prob. 38ECh. 13.2 - Evaluate the integral. (sec2ti+t(t2+1)3j+t2lntk)dtCh. 13.2 - Prob. 40ECh. 13.2 - Prob. 41ECh. 13.2 - Prob. 42ECh. 13.2 - Prob. 43ECh. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prove Formula 5 of Theorem 3.Ch. 13.2 - Prob. 46ECh. 13.2 - Prob. 47ECh. 13.2 - If u and v are the vector functions in Exercise...Ch. 13.2 - Prob. 49ECh. 13.2 - Prob. 50ECh. 13.2 - If r(t)=acost+bsint, where a and b are constant...Ch. 13.2 - Prob. 52ECh. 13.2 - Prob. 53ECh. 13.2 - Find an expression for ddt[u(t)(v(t)w(t))].Ch. 13.2 - Prob. 55ECh. 13.2 - Prob. 56ECh. 13.2 - Prob. 57ECh. 13.2 - Prob. 58ECh. 13.3 - Find the length of the curve....Ch. 13.3 - Prob. 2ECh. 13.3 - Prob. 3ECh. 13.3 - Prob. 4ECh. 13.3 - Find the length of the curve. r(t)=i+t2j+t3k,0t1Ch. 13.3 - Prob. 6ECh. 13.3 - Prob. 7ECh. 13.3 - Find the length of the curve correct of four...Ch. 13.3 - Prob. 9ECh. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Find, correct to four decimal places, the length...Ch. 13.3 - a Find the arc length function for the curve...Ch. 13.3 - a Find the arc length function for the curve...Ch. 13.3 - Prob. 15ECh. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - a Find the unit tangent and unit normal vectors...Ch. 13.3 - Prob. 18ECh. 13.3 - Prob. 19ECh. 13.3 - Prob. 20ECh. 13.3 - Use Theorem 10 to find the curvature. r(t)=t3j+t2kCh. 13.3 - Use Theorem 10 to find the curvature....Ch. 13.3 - Prob. 23ECh. 13.3 - Find the curvature of r(t)=t2,lnt,tlnt at the...Ch. 13.3 - Find the curvature of r(t)=t,t2,t3 at the point...Ch. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Use Formula 11 to find the curvature. y=x4Ch. 13.3 - Prob. 28ECh. 13.3 - Use Formula 11 to find the curvature. y=xexCh. 13.3 - Prob. 30ECh. 13.3 - Prob. 31ECh. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - a Is the curvature of the curve C shown in the...Ch. 13.3 - Prob. 34ECh. 13.3 - Prob. 35ECh. 13.3 - Prob. 36ECh. 13.3 - Prob. 37ECh. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Prob. 40ECh. 13.3 - Prob. 41ECh. 13.3 - Prob. 42ECh. 13.3 - Prob. 43ECh. 13.3 - Prob. 44ECh. 13.3 - Prob. 45ECh. 13.3 - Prob. 46ECh. 13.3 - Prob. 47ECh. 13.3 - Prob. 48ECh. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - Prob. 53ECh. 13.3 - Is there a point on the curve in Exercise 53 where...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Prob. 56ECh. 13.3 - Show that at every point on the curve...Ch. 13.3 - Prob. 58ECh. 13.3 - Prob. 59ECh. 13.3 - Prob. 60ECh. 13.3 - a Show that dB/ds is perpendicular to B. b Show...Ch. 13.3 - Prob. 62ECh. 13.3 - Use the Frenet-Serret formulas to prove each of...Ch. 13.3 - Show that the circular helix r(t)=acost,asint,bt,...Ch. 13.3 - Prob. 65ECh. 13.3 - Prob. 66ECh. 13.3 - Prob. 67ECh. 13.3 - Prob. 68ECh. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - The figure shows the path of a particle that moves...Ch. 13.4 - Prob. 3ECh. 13.4 - Prob. 4ECh. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 6ECh. 13.4 - Prob. 7ECh. 13.4 - Prob. 8ECh. 13.4 - Prob. 9ECh. 13.4 - Prob. 10ECh. 13.4 - Prob. 11ECh. 13.4 - Prob. 12ECh. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 15ECh. 13.4 - Prob. 16ECh. 13.4 - a Find the position vector of a particle that has...Ch. 13.4 - Prob. 18ECh. 13.4 - The position function of a particle is given by...Ch. 13.4 - Prob. 20ECh. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Prob. 24ECh. 13.4 - Prob. 25ECh. 13.4 - A projectile is fired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - A batter hits a baseball 3 ft above the ground...Ch. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Show that a projectile reaches three-quarters of...Ch. 13.4 - A ball is thrown eastward into the air from the...Ch. 13.4 - Prob. 32ECh. 13.4 - Water traveling along a straight portion of a...Ch. 13.4 - Prob. 34ECh. 13.4 - Prob. 35ECh. 13.4 - Prob. 36ECh. 13.4 - Prob. 37ECh. 13.4 - Prob. 38ECh. 13.4 - Prob. 39ECh. 13.4 - Prob. 40ECh. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Prob. 42ECh. 13.4 - The magnitude of the acceleration vector a is 10...Ch. 13.4 - Prob. 44ECh. 13.4 - The position function of a spaceship is...Ch. 13.4 - Prob. 46ECh. 13.R - Prob. 1CCCh. 13.R - Prob. 2CCCh. 13.R - Prob. 3CCCh. 13.R - Prob. 4CCCh. 13.R - Prob. 5CCCh. 13.R - Prob. 6CCCh. 13.R - Prob. 7CCCh. 13.R - Prob. 8CCCh. 13.R - Prob. 9CCCh. 13.R - Prob. 1TFQCh. 13.R - Prob. 2TFQCh. 13.R - Prob. 3TFQCh. 13.R - Prob. 4TFQCh. 13.R - Prob. 5TFQCh. 13.R - Prob. 6TFQCh. 13.R - Determine whether the statement is true or false....Ch. 13.R - Prob. 8TFQCh. 13.R - Prob. 9TFQCh. 13.R - Prob. 10TFQCh. 13.R - Prob. 11TFQCh. 13.R - Prob. 12TFQCh. 13.R - Prob. 13TFQCh. 13.R - Prob. 14TFQCh. 13.R - Prob. 1ECh. 13.R - Prob. 2ECh. 13.R - Prob. 3ECh. 13.R - Prob. 4ECh. 13.R - Prob. 5ECh. 13.R - Prob. 6ECh. 13.R - Prob. 7ECh. 13.R - Prob. 8ECh. 13.R - Prob. 9ECh. 13.R - Prob. 10ECh. 13.R - For the curve given by r(t)=sin3t,cos3t,sin2t,...Ch. 13.R - Find the curvature of the ellipse x=3cost,y=4sint...Ch. 13.R - Find the curvature of the curve y=x4 at the point...Ch. 13.R - Find an equation of the osculating circle of the...Ch. 13.R - Prob. 15ECh. 13.R - The figure shows the curve C traced by a particle...Ch. 13.R - A particle moves with position function...Ch. 13.R - Prob. 18ECh. 13.R - A particle starts at the origin with initial...Ch. 13.R - Prob. 20ECh. 13.R - A projectile is launched with an initial speed of...Ch. 13.R - Prob. 22ECh. 13.R - Prob. 23ECh. 13.R - In designing transfer curves to connect sections...Ch. 13.P - A particle P moves with constant angular speed ...Ch. 13.P - A circular curve of radius R on a highway is...Ch. 13.P - A projectile is fired from the origin with angle...Ch. 13.P - a A projectile is fired from the origin down an...Ch. 13.P - A ball rolls off a table with a speed of 2 ft/s....Ch. 13.P - Prob. 6PCh. 13.P - If a projectile is fired with angle of elevation ...Ch. 13.P - Prob. 8PCh. 13.P - Prob. 9P
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