Calculus Volume 3
Calculus Volume 3
16th Edition
ISBN: 9781938168079
Author: Gilbert Strang, Edwin Jed Herman
Publisher: OpenStax
bartleby

Concept explainers

Vector Arithmetic
Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magn…
Vector Calculus
Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.
Textbook Question
100%
Book Icon
Chapter 3.2, Problem 41E

Compute the derivatives of the vector-valued functions.

41. r ( t ) = t 3 i + 3 t 2 j + t 3 6 k

Expert Solution & Answer
Check Mark

Want to see the full answer?

Check out a sample textbook solution
Blurred answer
Previous
Chapter 3.1, Problem 40E
Next
Chapter 3.2, Problem 42E
Students have asked these similar questions
Find the domain of the vector valued function. r(t) = 3/(t-4) i + sqrt(3-t) j + ln|4-t| k.
q14
+ Dj +1 -² ₂2k -k. 2. Determine the numbers at which the vector-valued function R(t) = t²i + ln (t - 1)j

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...

Additional Math Textbook Solutions

Find more solutions based on key concepts
In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0...

Calculus Volume 1

Use basic integration formulas to compute the following antiderivatives. 209. dx2x

Calculus Volume 2

Use the following information to answer the next en exercises: A sample of 20 beads of lettuce was selected. As...

Introductory Statistics

The equivalent expression of x(y+z) by using the commutative property.

Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)

Suppose you toss one coin three times in a row and get heads, tails, heads (HTH). If you are interested in the ...

Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)

Michael says that the dark marbles in Figure 2.25 can’t represent 13 because “there are 5 marbles and 5 is more...

Mathematics for Elementary Teachers with Activities (5th Edition)

Knowledge Booster
Background pattern image
Math
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
SEE MORE TEXTBOOKS