Explanation Given: r ( t ) = 〈 t 2 2 , t 3 3 , t 〉 at t = 1 Formula used: N → ( t ) = v ( t ) r ' ' ( t ) − v ' ( t ) r ' ( t ) ‖ v ( t ) r ' ' ( t ) − v ' ( t ) r ' ( t ) ‖ r → ' ( t ) = d r → ( t ) d t v ( t ) = ‖ r → ' ( t ) ‖ r ' ' ( t ) = d 2 r → ( t ) d t 2 v ' ( t ) = d d t v → ( t ) Calculation: r → ( t ) = 〈 t 2 2 , t 3 3 , t 〉 r → ' ( t ) = d d t r → ( t ) = d d t 〈 t 2 2 , t 3 3 , t 〉 = d d t 〈 d d t t 2 2 , d d t t 3 3 , d d t t 〉 = 〈 1 2 × 2 t , 1 3 × 3 t 2 , 1 〉 = 〈 t , t 2 , 1 〉 ⋯ ⋯ ( 1 ) v ( t ) = ‖ r ' ( t ) ‖ = t 2 + t 4 + 1 ⋯ ⋯ ( 2 ) r → ' ' ( t ) = d d t ( r → ' ( t ) ) = d d t 〈 t , t 2 , 1 〉 = 〈 d d t t , d d t t 2 , d d t 1 〉 = 〈 1 , 2 t , 0 〉 ⋯ ⋯ ( 3 ) v ' ( t ) = 1 2 t 2 + t 4 + 1 × ( 2 t + 4 t 3 ) = 2 t 3 + t t 2 + t 4 + 1 ⋯ ⋯ ( 4 ) Substitute t = 1 in equation ( 1 ) , ( 2 ) , ( 3 ) , and ( 4 ) to find the value r → ' ( 1 ) , v → ( 1 ) , r → ' ' ( 1 ) and v → ' ' ( 1 ) r → ' ( 1 ) = 〈 1 , 1 , 1 〉 v ( 1 ) = 1 2 + 1 4 + 1

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These…

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are…

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) ge…

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.

Videos

Question

Chapter 14.4, Problem 49E

To determine

To calculate:

Normal vector $(\vec{N})$ at the indicated point.

$〈 \frac{t^{2}}{2}, \frac{t^{3}}{3}, t 〉 at t = 1$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.4, Problem 48E

Chapter 14.4, Problem 50E

Students have asked these similar questions

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z ≤ 3}. Calculate the integral f(x, y, z) dv. E

(12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.

Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Normal vector ( N → ) at the indicated point. 〈 t 2 2 , t 3 3 , t 〉 at t = 1

Solution Summary: The author explains how to calculate a normal vector (N ) at the indicated point.

Concept explainers

Videos

To calculate:

Normal vector $(\vec{N})$ at the indicated point.

$〈 \frac{t^{2}}{2}, \frac{t^{3}}{3}, t 〉 at t = 1$

Want to see the full answer?

Chapter 14 Solutions

Normal vector ( N → ) at the indicated point. 〈 t 2 2 , t 3 3 , t 〉 at t = 1

Solution Summary: The author explains how to calculate a normal vector (N ) at the indicated point.

Concept explainers

Videos

To calculate: Normal vector (N→) at the indicated point. 〈t22,t33,t〉 at t=1

Want to see the full answer?

Chapter 14 Solutions

To calculate:

Normal vector $(\vec{N})$ at the indicated point.

$〈 \frac{t^{2}}{2}, \frac{t^{3}}{3}, t 〉 at t = 1$