Calculus Volume 3
16th Edition
ISBN: 9781938168079
Author: Gilbert Strang, Edwin Jed Herman
Publisher: OpenStax
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Chapter 3.3, Problem 116E
Given
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A function is defined on the interval (-π/2,π/2) by this multipart rule:
if -π/2 < x < 0
f(x) =
a
if x=0
31-tan x
+31-cot x
if 0 < x < π/2
Here, a and b are constants. Find a and b so that the function f(x) is continuous at x=0.
a=
b= 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
f(x) = (x + 4x4) 5,
a = -1
lim f(x)
X--1
=
lim
x+4x
X--1
lim
X-1
4
x+4x
5
))"
5
))
by the power law
by the sum law
lim (x) + lim
X--1
4
4x
X-1
-(0,00+(
Find f(-1).
f(-1)=243
lim (x) +
-1 +4
35
4 ([
)
lim (x4)
5
x-1
Thus, by the definition of continuity, f is continuous at a = -1.
by the multiple constant law
by the direct substitution property
Chapter 3 Solutions
Calculus Volume 3
Ch. 3.1 - Give the component functions x=f(t) and y=g(t) for...Ch. 3.1 - Given r(t)=3secti+2tantj , find the following...Ch. 3.1 - Sketch the curve of the vector-valued function...Ch. 3.1 - Evaluate limt0eti+sinttj+etk .Ch. 3.1 - Given the vector-valued function r(t)=cost,sint ,...Ch. 3.1 - Given the vector-valued function r(t)=t,t2+1 ,...Ch. 3.1 - Let r(t)=eti+sintj+lntk . Find the following...Ch. 3.1 - Find the limit of the following vector-valued...Ch. 3.1 - Find the limit of the following vector-valued...Ch. 3.1 - Find the limit of the following vector-valued...
Ch. 3.1 - Find the limit of the following vector-valued...Ch. 3.1 - Find the limit of the following vector-valued...Ch. 3.1 - Find the limit of the following vector-valued...Ch. 3.1 - Find the limit of the following vector-valued...Ch. 3.1 - Find the domain of the vector-valued functions....Ch. 3.1 - Find the domain of the vector-valued functions....Ch. 3.1 - Find the domain of the vector-valued functions....Ch. 3.1 - Let r(t)=cost,t,sint and use it to answer the...Ch. 3.1 - Let r(t)=cost,t,sint and use it to answer the...Ch. 3.1 - Let r(t)=cost,t,sint and use it to answer the...Ch. 3.1 - Let r(t)=cost,t,sint and use it to answer the...Ch. 3.1 - Eliminate the parameter t, write the equation in...Ch. 3.1 - Eliminate the parameter t, write the equation in...Ch. 3.1 - Eliminate the parameter t, write the equation in...Ch. 3.1 - Eliminate the parameter t, write the equation in...Ch. 3.1 - Eliminate the parameter t, write the equation in...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - Use a graphing utility to sketch each of the...Ch. 3.1 - [T] Let r(t)=costi+sintj+0.3sin(2t)k . Use...Ch. 3.1 - [T] Use the result of the preceding problem to...Ch. 3.1 - Use the results If the preceding two problems to...Ch. 3.1 - a. Graph the curve...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - Compute the derivatives of the vector-valued...Ch. 3.2 - For the following problems, find a tangent vector...Ch. 3.2 - For the following problems, find a tangent vector...Ch. 3.2 - For the following problems, find a tangent vector...Ch. 3.2 - For the following problems, find a tangent vector...Ch. 3.2 - Find the unit tangent vector for the following...Ch. 3.2 - Find the unit tangent vector for the following...Ch. 3.2 - Find the unit tangent vector for the following...Ch. 3.2 - Find the unit tangent vector for the following...Ch. 3.2 - Find the following. 59. ddt[r(t2)]Ch. 3.2 - Find the following. 60. ddt[t2.s(t)]Ch. 3.2 - Find the following. 61. ddt[r(t).s(t)]Ch. 3.2 - Compute the first, second, and third derivatives...Ch. 3.2 - Find r(t)r(t) for r(t)=3t5i+5tj+2t2k .Ch. 3.2 - The acceleration function, initial velocity, and...Ch. 3.2 - The position vector of a particle is...Ch. 3.2 - Find the velocity and the speed of a panicle with...Ch. 3.2 - Find the velocity function and show that v(t) is...Ch. 3.2 - Show that the speed of the particle is...Ch. 3.2 - Evaluate ddt[u(t)u(t)] given u(t)=t2i2tj+k .Ch. 3.2 - Find the antiderivative of...Ch. 3.2 - Evaluate 03ti+t2jdt .Ch. 3.2 - An object starts from nest at point P(1,2,0) and...Ch. 3.2 - Show that if the speed 0f a particle traveling...Ch. 3.2 - Given r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k , find...Ch. 3.2 - Given r(t)=t+cost,tsint , find the velocity and...Ch. 3.2 - Find the velocity vector for the function...Ch. 3.2 - Find the equation of the tangent line to the curve...Ch. 3.2 - Describe and sketch the curve represented by the...Ch. 3.2 - Locate the highest point on the curve r(t)=6t,6tt2...Ch. 3.2 - The position vector for a particle is...Ch. 3.2 - The position vector for a particle is...Ch. 3.2 - The position vector for a particle is...Ch. 3.2 - A particle travels along the path of a helix with...Ch. 3.2 - A particle travels along the path of a helix with...Ch. 3.2 - A particle travels along the path of a helix with...Ch. 3.2 - A particle travels along the path of a helix with...Ch. 3.2 - A particle travels along the path of an ellipse...Ch. 3.2 - A particle travels along the path of an ellipse...Ch. 3.2 - A particle travels along the path of an ellipse...Ch. 3.2 - Given the vector-valued function r(t)=tant,sect,0...Ch. 3.2 - Given the vector-valued function r(t)=tant,sect,0...Ch. 3.2 - Given the vector-valued function r(t)=tant,sect,0...Ch. 3.2 - Find the minimum speed of a particle traveling...Ch. 3.2 - Given r(t)=ti+2sintj+2costk and...Ch. 3.2 - Given r(t)=ti+2sintj+2costk and...Ch. 3.2 - Now, use the product rule for the derivative of...Ch. 3.2 - Find the unit tangent vector T(t) for the...Ch. 3.2 - Find the unit tangent vector T(t) for the...Ch. 3.2 - Find the unit tangent vector T(t) for the...Ch. 3.2 - Evaluate the following integrals: 100. ( e...Ch. 3.2 - Evaluate the following integrals: 101. 01r(t)dt ,...Ch. 3.3 - Find the arc length of the curve on the given...Ch. 3.3 - Find the arc length of the curve on the given...Ch. 3.3 - Find the arc length of the curve on the given...Ch. 3.3 - Find the arc length of the curve on the given...Ch. 3.3 - r(t)=etcost,etsint over the interval [0,2] . Here...Ch. 3.3 - Find the length of one turn of the helix given by...Ch. 3.3 - Find the arc length of the vector-valued function...Ch. 3.3 - A particle travels in a circle with the equation...Ch. 3.3 - Set up an integral to find the circumference of...Ch. 3.3 - Find the length of the curve r(t)=2t,et,et over...Ch. 3.3 - Find the length of the curve r(t)=2sint,5t,2cost...Ch. 3.3 - The position function for a particle is...Ch. 3.3 - Given r(t)=acos(t)i+bsin(t)j , find the binormal...Ch. 3.3 - Given r(t)=2et,etcost,etsint , determine the...Ch. 3.3 - Given r(t)=2et,etcost,etsint , determine the unit...Ch. 3.3 - Given r(t)=2et,etcost,etsint , find the unit...Ch. 3.3 - Given r(t)=2et,etcost,etsint , find the unit...Ch. 3.3 - Given r(t)=ti+t2j+tk . find the unit tangent...Ch. 3.3 - Find the unit tangent vector T(t) and unit normal...Ch. 3.3 - Find the unit tangent vector T(t) for...Ch. 3.3 - Find the principal normal vector to the curve...Ch. 3.3 - Find T(t) for the curve r(t)=(t34t)i+(5t22)j .Ch. 3.3 - Find N(t) for the curve r(t)=(t34t)i+(5t22)j .Ch. 3.3 - Find the unit normal vector N(t) for...Ch. 3.3 - Find the unit tangent vector T(t) for...Ch. 3.3 - Find the arc length function s(t) for the line...Ch. 3.3 - Parameterize the helix r(t)=costi+sintj+tk using...Ch. 3.3 - Parameterize the curve using the arc-length...Ch. 3.3 - Find the curvature of the curve r(t)=5costi+4sintj...Ch. 3.3 - Find the x-coordinate at which the curvature of...Ch. 3.3 - Find the curvature of the curve r(t)=5costi+5sintj...Ch. 3.3 - Find the curvature k for the curve y=x14x2 at the...Ch. 3.3 - Find the curvature k for the curve y=13x3 at the...Ch. 3.3 - Find the curvature k of the curve r(t)=ti+6t2j+4tk...Ch. 3.3 - Find the mature of r(t)=2sint,5t,2cost .Ch. 3.3 - Find the curvature of r(t)=2ti+etj+etk at point...Ch. 3.3 - At what point does the curve y=ex have maximum...Ch. 3.3 - What happens to the curvature as x on for the...Ch. 3.3 - Find the point of maximum curvature on the curve...Ch. 3.3 - Find the equations of the normal plane and the...Ch. 3.3 - Find equations of the osculating circles of the...Ch. 3.3 - Find the equation for the osculating plane at...Ch. 3.3 - Find the radius of curvature of 6y=x3 at the point...Ch. 3.3 - Find the curvature at each point (x,y) on the...Ch. 3.3 - Calculate the mature of the circular helix...Ch. 3.3 - Find the radius of curvature of y=ln(x+1) at point...Ch. 3.3 - Find the radius of curvature of the hyperbola xy=1...Ch. 3.3 - A particle moves along the plane curve C described...Ch. 3.3 - A particle moves along the plane curve C described...Ch. 3.3 - A particle moves along the plane curve C described...Ch. 3.3 - The surface of a large cup is formed by revolving...Ch. 3.3 - The surface of a large cup is formed by revolving...Ch. 3.3 - The surface of a large cup is formed by revolving...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - How fast can a racecar travel through a circular...Ch. 3.4 - Given r(t)=(3t22)i+(2tsin(t))j , find the velocity...Ch. 3.4 - Given r(t)=(3t22)i+(2tsin(t))j , find the...Ch. 3.4 - Given the following position functions, find the...Ch. 3.4 - Given the following position functions, find the...Ch. 3.4 - Given the following position functions, find the...Ch. 3.4 - Find the velocity, acceleration, and speed of a...Ch. 3.4 - Find the velocity, acceleration, and speed of a...Ch. 3.4 - Find the velocity, acceleration, and speed of a...Ch. 3.4 - The position function of an object is given by...Ch. 3.4 - Let r(t)=rcosh(t)i+rsinh(wt)j . Find the velocity...Ch. 3.4 - Consider the motion of a point on the...Ch. 3.4 - A person on a hang glider is spiraling upward as a...Ch. 3.4 - A person on a hang glider is spiraling upward as a...Ch. 3.4 - A person on a hang glider is spiraling upward as a...Ch. 3.4 - Given that r(t)=e5tsint,e5tcost,4e5t is the...Ch. 3.4 - Given that r(t)=e5tsint,e5tcost,4e5t is the...Ch. 3.4 - Given that r(t)=e5tsint,e5tcost,4e5t is the...Ch. 3.4 - Given that r(t)=e5tsint,e5tcost,4e5t is the...Ch. 3.4 - A projectile is shot in the air from ground level...Ch. 3.4 - A projectile is shot in the air from ground level...Ch. 3.4 - A projectile is shot in the air from ground level...Ch. 3.4 - A projectile is shot in the air from ground level...Ch. 3.4 - A projectile is shot in the air from ground level...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - A projectile is fired at a height of 1.5m above...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - For each of the following problems, find the...Ch. 3.4 - All automobile that weighs 2700lb makes a turn on...Ch. 3.4 - Using Kepler’s laws, it can be shown that v0=2GMr0...Ch. 3.4 - Find the lime in years it takes the dwarf planet...Ch. 3.4 - Suppose that the position function for an object...Ch. 3.4 - Suppose that the position function for an object...Ch. 3.4 - Suppose that the position function for an object...Ch. 3 - True or False? Justify your answer with a proof or...Ch. 3 - True or False? Justify your answer with a proof or...Ch. 3 - True or False? Justify your answer with a proof or...Ch. 3 - True or False? Justify your answer with a proof or...Ch. 3 - Find the domains of the vector-valued functions....Ch. 3 - Find the domains of the vector-valued functions....Ch. 3 - Sketch the tunes. for the following vector...Ch. 3 - Sketch the tunes. for the following vector...Ch. 3 - Find a vector function that describes the...Ch. 3 - Find a vector function that describes the...Ch. 3 - Find the derivatives of u(t),u(t),u(t)u(t) ,...Ch. 3 - Find the derivatives of u(t),u(t),u(t)u(t) ,...Ch. 3 - Evaluate the following integrals. 214. (tan(...Ch. 3 - Evaluate the following integrals. 215. 14(t)dt ,...Ch. 3 - Find the length for the following curves. 216....Ch. 3 - Find the length for the following curves. 217....Ch. 3 - Reparametrize the following functions with respect...Ch. 3 - Reparametrize the following functions with respect...Ch. 3 - Find the curvature for the following vector...Ch. 3 - Find the curvature for the following vector...Ch. 3 - Find the curvature for the following vector...Ch. 3 - Find the curvature for the following vector...Ch. 3 - Find the curvature for the following vector...Ch. 3 - Find the curvature for the following vector...Ch. 3 - The following problems consider launching a...Ch. 3 - The following problems consider launching a...Ch. 3 - The following problems consider launching a...
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