Thomas' Calculus Format: Unbound (saleable) With Access Card
Thomas' Calculus Format: Unbound (saleable) With Access Card
14th Edition
ISBN: 9780134768762
Author: Hass, Joel R.^heil, Christopher D.^weir, Maurice D.
Publisher: Prentice Hall
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Chapter 13, Problem 1GYR
To determine

Mention the rules for differentiating and integrating vector functions with some examples.

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Rules for differentiating vector functions:

Consider,

u and v is the differentiable vector functions of t,

c is scalar,

C is a constant vector, and

f is differentiable scalar function.

1. Constant function rule:

ddtC=0

2. Sum rule:

ddt[u(t)+v(t)]=u'(t)+v'(t)

3. Difference rule:

ddt[u(t)v(t)]=u'(t)v'(t)

4. Scalar multiple rule:

ddt[cu(t)]=cu'(t)

5. Chain rule:

ddt[u(f(t))]=cu'(t)

6. Dot product rule:

ddt[u(t)v(t)]=u'(t)v(t)+u(t)v'(t)

7. Cross product rule:

ddt[u(t)×v(t)]=u'(t)×v(t)+u(t)×v'(t)

For example:

Consider the position of a particle in the xy-plane r(t)=(3t+1)i+3tj+t2k. Find the angle between the velocity and acceleration vectors at time t=0.

The position function is,

r(t)=(3t+1),3t,t2

The expression for velocity of a particle is,

v=drdt

Substitute (3t+1),3t,t2 for r in above equation.

v=ddt((3t+1),3t,t2)=3,3,2t

At t=0, the velocity of the particle is,

v(0)=3,3,2(0)=3,3,0

The magnitude of the velocity v is,

|v(0)|=32+(3)2=9+3=12

The expression for acceleration of a particle.

a=dvdt

Substitute 3,3,2t for v in above equation.

a=ddt(3,3,2t)=0,0,2

At t=0, the acceleration of the particle is,

a(0)=0,0,2

The magnitude of the acceleration a is,

|a(0)|=22=4=2

The expression to find the angle between two vectors a and b.

θ=cos1(ab|a||b|)

The expression to find the angle between two vectors a and b at time t=0.

θ=cos1(v(0)a(0)|v(0)||a(0)|)

Substitute 0,0,2 for a(0), 3,3,0 for v(0), 12 for |v(0)|, and 2 for |a(0)| in above equation as follows.

θ=cos1(3,3,00,0,2(12)(2))=cos1(0)

The above equation becomes,

θ=π2

Therefore, the angle between the velocity and acceleration vectors at given time is θ=π2.

Rules for integrating vector functions:

The indefinite integral of r with respect to t is the set of all antiderivatives of r. It is represented by r(t)dt. Consider if R is antiderivative of r, then

r(t)dt=R(t)+C

For example:

Integrate a vector function [(2cost)i+j2tk]dt.

[(2cost)i+j2tk]dt=(2costdt)i+(dt)j(2tdt)k=(2sint+C1)i+(t+C2)j(2t22+C3)k=(2sint)i+tj2t2k+C1i+C2jC3k=(2sint)i+tj2t2k+C{C=C1i+C2jC3k}

Thus, the rules for differentiating and integrating vector functions is explained with an examples.

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Chapter 13 Solutions

Thomas' Calculus Format: Unbound (saleable) With Access Card

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 - In Exercises 19–22, r(t) is the position of a...Ch. 13.1 - Prob. 21ECh. 13.1 - Prob. 22ECh. 13.1 - Prob. 23ECh. 13.1 - Tangents to Curves As mentioned in the text, the...Ch. 13.1 - Tangents to Curves As mentioned in the text, the...Ch. 13.1 - Tangents to Curves As mentioned in the text, the...Ch. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - Prob. 29ECh. 13.1 - In Exercises 27-30, find the value(s) of t so that...Ch. 13.1 - Prob. 31ECh. 13.1 - Prob. 32ECh. 13.1 - Prob. 33ECh. 13.1 - Prob. 34ECh. 13.1 - Prob. 35ECh. 13.1 - Prob. 36ECh. 13.1 - Prob. 37ECh. 13.1 - Prob. 38ECh. 13.1 - Prob. 39ECh. 13.1 - Prob. 40ECh. 13.1 - Prob. 41ECh. 13.1 - Prob. 42ECh. 13.1 - Prob. 43ECh. 13.1 - Prob. 44ECh. 13.1 - Prob. 45ECh. 13.1 - Prob. 46ECh. 13.1 - Prob. 47ECh. 13.1 - Prob. 48ECh. 13.2 - Evaluate the integrals in Exercises 1–10. 1. Ch. 13.2 - Prob. 2ECh. 13.2 - Prob. 3ECh. 13.2 - Evaluate the integrals in Exercises 1–10. 4. Ch. 13.2 - Prob. 5ECh. 13.2 - Prob. 6ECh. 13.2 - Prob. 7ECh. 13.2 - Evaluate the integrals in Exercises 1–10. 8. Ch. 13.2 - Prob. 9ECh. 13.2 - Prob. 10ECh. 13.2 - Prob. 11ECh. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Prob. 13ECh. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Prob. 15ECh. 13.2 - Prob. 16ECh. 13.2 - Prob. 17ECh. 13.2 - Prob. 18ECh. 13.2 - Prob. 19ECh. 13.2 - Solve the initial value problems in Exercises...Ch. 13.2 - Prob. 21ECh. 13.2 - Prob. 22ECh. 13.2 - Prob. 23ECh. 13.2 - Prob. 24ECh. 13.2 - Prob. 25ECh. 13.2 - Throwing a baseball A baseball is thrown from the...Ch. 13.2 - Prob. 27ECh. 13.2 - Beaming electrons An electron in a TV tube is...Ch. 13.2 - Prob. 29ECh. 13.2 - Prob. 30ECh. 13.2 - Prob. 31ECh. 13.2 - Prob. 32ECh. 13.2 - Prob. 33ECh. 13.2 - Prob. 34ECh. 13.2 - Launching downhill An ideal projectile is...Ch. 13.2 - Prob. 36ECh. 13.2 - Prob. 37ECh. 13.2 - Prob. 38ECh. 13.2 - Prob. 39ECh. 13.2 - Prob. 40ECh. 13.2 - Prob. 41ECh. 13.2 - Hitting a baseball with linear drag Consider the...Ch. 13.2 - Prob. 43ECh. 13.2 - Prob. 44ECh. 13.2 - Prob. 45ECh. 13.2 - Prob. 46ECh. 13.2 - Hitting a baseball with linear drag under a wind...Ch. 13.2 - Prob. 48ECh. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 13.3 - Prob. 9ECh. 13.3 - Prob. 10ECh. 13.3 - Prob. 11ECh. 13.3 - In Exercises 11–14, find the arc length parameter...Ch. 13.3 - Prob. 13ECh. 13.3 - Prob. 14ECh. 13.3 - Prob. 15ECh. 13.3 - Length of helix The length of the turn of the...Ch. 13.3 - Length is independent of parametrization To...Ch. 13.3 - Prob. 19ECh. 13.3 - (Continuation of Exercise 19.) Find the unit...Ch. 13.3 - Distance along a line Show that if u is a unit...Ch. 13.3 - Prob. 22ECh. 13.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 13.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 13.4 - Prob. 3ECh. 13.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 13.4 - A formula for the curvature of the graph of a...Ch. 13.4 - A formula for the curvature of a parametrized...Ch. 13.4 - Normals to plane curves Show that n(t) = −g′(t)i...Ch. 13.4 - (Continuation of Exercise 7.) Use the method of...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Prob. 12ECh. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Find T, N, and κ for the space curves in Exercises...Ch. 13.4 - Show that the parabola , has its largest curvature...Ch. 13.4 - Prob. 18ECh. 13.4 - Prob. 19ECh. 13.4 - Prob. 20ECh. 13.4 - Find an equation for the circle of curvature of...Ch. 13.4 - Find an equation for the circle of curvature of...Ch. 13.4 - Prob. 23ECh. 13.4 - The formula derived in Exercise 5, expresses the...Ch. 13.4 - Prob. 25ECh. 13.4 - Prob. 26ECh. 13.4 - Prob. 27ECh. 13.4 - Prob. 28ECh. 13.4 - Osculating circle Show that the center of the...Ch. 13.4 - Prob. 30ECh. 13.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 13.5 - Prob. 2ECh. 13.5 - Prob. 3ECh. 13.5 - Prob. 4ECh. 13.5 - Prob. 5ECh. 13.5 - Prob. 6ECh. 13.5 - Prob. 7ECh. 13.5 - In Exercises 7 and 8, find r, T, N, and B at the...Ch. 13.5 - Prob. 9ECh. 13.5 - Prob. 10ECh. 13.5 - Prob. 11ECh. 13.5 - Prob. 12ECh. 13.5 - Prob. 13ECh. 13.5 - Prob. 14ECh. 13.5 - Prob. 15ECh. 13.5 - Prob. 16ECh. 13.5 - Prob. 17ECh. 13.5 - Prob. 18ECh. 13.5 - Prob. 19ECh. 13.5 - Prob. 20ECh. 13.5 - Prob. 21ECh. 13.5 - Prob. 22ECh. 13.5 - Prob. 23ECh. 13.5 - Prob. 24ECh. 13.5 - Prob. 25ECh. 13.5 - Prob. 26ECh. 13.6 - Prob. 1ECh. 13.6 - Prob. 2ECh. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - Prob. 4ECh. 13.6 - Prob. 5ECh. 13.6 - In Exercises 1–7, find the velocity and...Ch. 13.6 - Prob. 7ECh. 13.6 - Type of orbit For what values of v0 in Equation...Ch. 13.6 - Prob. 9ECh. 13.6 - Prob. 10ECh. 13.6 - Prob. 11ECh. 13.6 - Prob. 12ECh. 13.6 - Prob. 13ECh. 13.6 - Prob. 14ECh. 13.6 - Prob. 15ECh. 13.6 - Prob. 16ECh. 13.6 - Prob. 17ECh. 13.6 - Prob. 18ECh. 13 - Prob. 1GYRCh. 13 - Prob. 2GYRCh. 13 - Prob. 3GYRCh. 13 - Prob. 4GYRCh. 13 - Prob. 5GYRCh. 13 - Prob. 6GYRCh. 13 - Prob. 7GYRCh. 13 - Prob. 8GYRCh. 13 - Prob. 9GYRCh. 13 - Prob. 10GYRCh. 13 - Prob. 11GYRCh. 13 - Prob. 12GYRCh. 13 - Prob. 13GYRCh. 13 - Prob. 1PECh. 13 - Prob. 2PECh. 13 - Prob. 3PECh. 13 - Prob. 4PECh. 13 - Prob. 5PECh. 13 - Prob. 6PECh. 13 - Prob. 7PECh. 13 - Prob. 8PECh. 13 - Prob. 9PECh. 13 - Prob. 10PECh. 13 - Prob. 11PECh. 13 - Prob. 12PECh. 13 - Prob. 13PECh. 13 - Prob. 14PECh. 13 - Prob. 15PECh. 13 - Prob. 16PECh. 13 - Prob. 17PECh. 13 - Prob. 18PECh. 13 - Prob. 19PECh. 13 - Prob. 20PECh. 13 - Prob. 21PECh. 13 - Prob. 22PECh. 13 - Prob. 23PECh. 13 - Prob. 24PECh. 13 - Prob. 25PECh. 13 - Prob. 26PECh. 13 - Prob. 27PECh. 13 - Prob. 28PECh. 13 - Prob. 29PECh. 13 - Prob. 30PECh. 13 - Prob. 31PECh. 13 - Prob. 32PECh. 13 - Prob. 1AAECh. 13 - Prob. 2AAECh. 13 - Prob. 3AAECh. 13 - Prob. 4AAECh. 13 - Prob. 5AAECh. 13 - Prob. 6AAECh. 13 - Prob. 7AAECh. 13 - Prob. 8AAECh. 13 - Prob. 9AAE
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