Explanation Formula used: d s = ‖ r ′ ( t ) ‖ d t ∫ C f ( x , y , z ) d s = ∫ t = a b f ( r ( t ) ) ‖ r ′ ( t ) ‖ d t Calculations: Given f ( x , y , z ) = x + 2 y + z and r ( t ) = 〈 cos t , sin t , t 〉 for 0 ≤ t ≤ π 2 Since r ( t ) = 〈 cos t , sin t , t 〉 , so x = cos t , y = sin t , z = t Thus, f ( x , y , z ) = x + 2 y + z f ( r ( t ) ) = cos t + 2 sin t + t Now, since r ( t ) = 〈 cos t , sin t , t 〉 , so r ′ ( t ) = 〈 − sin t , cos t , 1 〉 Hence, ‖ r ′ ( t ) ‖ = ( − sin t ) 2 + ( cos t ) 2 +

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic…

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along…

Triple Integral

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Chapter 17, Problem 31CRE

To determine

To compute the line integral:

$f (x, y, z) = x + 2 y + z$ , the helix $r (t) = 〈 \cos t, \sin t, t 〉 for 0 \leq t \leq \frac{π}{2}$

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Chapter 17, Problem 30CRE

Chapter 17, Problem 32CRE

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Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties to find lim→-4 1 [2h (x) — h(x) + 7 f(x)] : - h(x)+7f(x) 3 O DNE

17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t). (a) How much of the slope field can you sketch from this information? [Hint: Note that the differential equation depends only on t.] (b) What can you say about the solu- tion with y(0) = 2? (For example, can you sketch the graph of this so- lution?) y(0) = 1 y AN

(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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To compute the line integral: f ( x , y , z ) = x + 2 y + z , the helix r ( t ) = 〈 cos t , sin t , t 〉 for 0 ≤ t ≤ π 2

Solution Summary: The author explains how the helix r (t) = cos t, sin

Concept explainers

Videos

To compute the line integral:

$f (x, y, z) = x + 2 y + z$ , the helix $r (t) = 〈 \cos t, \sin t, t 〉 for 0 \leq t \leq \frac{π}{2}$

Want to see the full answer?

Chapter 17 Solutions

To compute the line integral: f ( x , y , z ) = x + 2 y + z , the helix r ( t ) = 〈 cos t , sin t , t 〉 for 0 ≤ t ≤ π 2

Solution Summary: The author explains how the helix r (t) = cos t, sin

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To compute the line integral: f(x,y,z)=x+2y+z, the helix r(t)=〈cost,sint,t〉 for 0≤t≤π2

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Chapter 17 Solutions

To compute the line integral:

$f (x, y, z) = x + 2 y + z$ , the helix $r (t) = 〈 \cos t, \sin t, t 〉 for 0 \leq t \leq \frac{π}{2}$