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A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds. t (s) v (m/s) t (s) v (m/s) 0 0 3.0 10.51 0.5 4.67 3.5 10.67 1.0 7.34 4.0 10.76 1.5 8.86 4.5 10.81 2.0 9.73 5.0 10.81 2.5 10.22

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Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636
Chapter 4, Problem 48RE
Textbook Problem
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A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds.

t (s) v (m/s) t(s) v (m/s)
0 0 3.0 10.51
0.5 4.67 3.5 10.67
1.0 7.34 4.0 10.76
1.5 8.86 4.5 10.81
2.0 9.73 5.0 10.81
2.5 10.22    

To determine

To calculate: The distance covered by the runner during 5 seconds with Midpoint rule.

Explanation of Solution

Given information:

Consider number of intervals n=5.

The interval varies from i=1,2,....,n.

The region lies between t=0 and t=5sec.

Apply midpoint rule:

abf(x)dxi=1nf(x¯i)Δx=Δx(f(x¯1)+.......+f(x¯n))

Calculate the width of the interval using the formula.

Δx=ban (1)

Calculate the midpoint using the formula.

x¯i=12(xi1+xi) (2)

Here, width of the subintervals is Δx, the upper limit is b, the lower limit is a and the midpoint of (xi1,xi) is x¯i.

To find the width of the subintervals using Equation (1) as shown below.

Substitute 5 for n, 5 for b and 0 for a in Equation (1).

Δx=505=1

The expression to find the distance of the runner using midpoint rule as shown below

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Chapter 4 Solutions

Single Variable Calculus
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Ch. 4.1 - Some computer algebra systems have commands that...Ch. 4.1 - (a) If f(x)=x/(x+2), 1 x 4, use the commands...Ch. 4.1 - The speed of a runner increased steadily during...Ch. 4.1 - The table shows speedometer readings at 10-second...Ch. 4.1 - Oil leaked from a tank at a rate of r(t) liters...Ch. 4.1 - When we estimate distances from velocity data, it...Ch. 4.1 - The velocity graph of a braking car is shown. 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(arctanx2)x2+1dxCh. 4.5 - Evaluate the integral. 74. xx2+4dxCh. 4.5 - Evaluate the integral. 75. 1+x1+x2dxCh. 4.5 - Evaluate the integral. 76. sin(lnx)xdxCh. 4.5 - Evaluate the integral. 77. sin2x1+cos2xdxCh. 4.5 - Evaluate the integral. 78. sinx1+cos2xdxCh. 4.5 - Evaluate the integral. 79. cotxdxCh. 4.5 - Evaluate the integral. 80. x1+x4dxCh. 4.5 - Evaluate the integral. 81. ee4dxxlnxCh. 4.5 - Evaluate the integral. 82. 01xex2dxCh. 4.5 - Evaluate the integral. 83. 01ez+1ez+zdzCh. 4.5 - Evaluate the integral. 84. 01(x1)e(x1)2dxCh. 4.5 - Use Exercise 64 to evaluate the integral...Ch. 4 - (a) Write an expression for a Riemann sum of a...Ch. 4 - (a) Write the definition of the definite integral...Ch. 4 - State the Midpoint Rule.Ch. 4 - State both parts of the Fundamental Theorem of...Ch. 4 - (a) State the Net Change Theorem. (b) If r(t) is...Ch. 4 - Suppose a particle moves back and forth along a...Ch. 4 - (a) Explain the meaning of the indefinite integral...Ch. 4 - Explain exactly what is meant by the statement...Ch. 4 - State the Substitution Rule. In practice, how do...Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Use the given graph of f to find the Riemann sum...Ch. 4 - (a) Evaluate the Riemann sum for f(x)=x2x0x2 with...Ch. 4 - Evaluate 01(x+1x2)dx by interpreting it in terms...Ch. 4 - Express limni=1nsinxix as a definite integral on...Ch. 4 - If 06f(x)dx=10 and04f(x)dx=7, find46f(x)dx.Ch. 4 - (a) Write 15(x+2x5)dx as a limit of Riemann sums,...Ch. 4 - The figure shows the graphs of f,f, and 0xf(t)dt....Ch. 4 - Evaluate: (a) 0/2ddx(sinx2cosx3)dx (b)...Ch. 4 - The graph of f consists of the three line segments...Ch. 4 - If f is the function in Exercise 9, find g(4).Ch. 4 - Evaluate the integral, if it exists. 11....Ch. 4 - Evaluate the integral, if it exists. 12....Ch. 4 - Evaluate the integral, if it exists. 13. 01(1x9)dxCh. 4 - Evaluate the integral, if it exists. 14. 01(1x)9dxCh. 4 - Evaluate the integral, if it exists. 15. 19u2u2uduCh. 4 - Evaluate the integral, if it exists. 16....Ch. 4 - Evaluate the integral, if it exists. 17....Ch. 4 - Evaluate the integral, if it exists. 18....Ch. 4 - Evaluate the integral, if it exists. 19. 15dt(t4)2Ch. 4 - Evaluate the integral, if it exists. 20....Ch. 4 - Evaluate the integral, if it exists. 21....Ch. 4 - Evaluate the integral, if it exists. 22....Ch. 4 - Evaluate the integral, if it exists. 23....Ch. 4 - Evaluate the integral, if it exists. 24....Ch. 4 - Evaluate the integral, if it exists. 25....Ch. 4 - Evaluate the integral, if it exists. 26....Ch. 4 - Evaluate the integral, if it exists. 27....Ch. 4 - Evaluate the integral, if it exists. 28....Ch. 4 - Evaluate the integral, if it exists. 29. 03|x24|dxCh. 4 - Evaluate the integral, if it exists. 30. 04|x1|dxCh. 4 - Evaluate the indefinite integral. Illustrate and...Ch. 4 - Evaluate the indefinite integral. Illustrate and...Ch. 4 - Use a graph to give a rough estimate of the area...Ch. 4 - Graph the function f(x) = cos2x sin x and use the...Ch. 4 - Find the derivative of the function. 35....Ch. 4 - Find the derivative of the function. 36....Ch. 4 - Find the derivative of the function. 37....Ch. 4 - Find the derivative of the function. 38....Ch. 4 - Find the derivative of the function. 39.y=xxcosdCh. 4 - Find the derivative of the function. 40....Ch. 4 - Use Property 8 of integrals to estimate the value...Ch. 4 - Use Property 8 of integrals to estimate the value...Ch. 4 - Use the properties of integrals to verify the...Ch. 4 - Use the properties of integrals to verify the...Ch. 4 - Use the Midpoint Rule with n = 6 to approximate...Ch. 4 - A particle moves along a line with velocity...Ch. 4 - Let r(t) be the rate at which the worlds oil is...Ch. 4 - A radar gun was used to record the speed of a...Ch. 4 - A population of honeybees increased at a rate of...Ch. 4 - Let f(x)={x1if3x01x2if0x1 Evaluate 31f(x)dx by...Ch. 4 - If f is continuous and 02f(x)dx=6, evaluate...Ch. 4 - The Fresnel function S(x)=0xsin(12t2)dt was...Ch. 4 - If f is a continuous function such that...Ch. 4 - Find a function f and a value of the constant a...Ch. 4 - If f' is continuous on [a, b], show that...Ch. 4 - Find limh01h22+h1+t3dtCh. 4 - If f is continuous on [0, 1], prove that...Ch. 4 - Evaluate limn1n[(1n)9+(2n)9+(3n)9+...+(nn)9]Ch. 4 - If xsinx=0x2f(t)dt, where f is a continuous...Ch. 4 - Find the minimum value of the area of the region...Ch. 4 - If f is a differentiable function such that f(x)...Ch. 4 - (a) Graph several members of the family of...Ch. 4 - If f(x)0g(x)11+t3dt, where...Ch. 4 - If f(x)=0xx2sin(t2)dt, find f(x).Ch. 4 - Find the interval [a, b] for which the value of...Ch. 4 - Use an integrai to estimate th sum i=110000i.Ch. 4 - (a) Evaluate 0nxdx, where n is a positive integer....Ch. 4 - Find d2dx20x(1sint1+u4du)dt.Ch. 4 - Suppose the coefficients of the cubic polynomial...Ch. 4 - A circular disk of radius r is used in an...Ch. 4 - Prove that if f is continuous, then...Ch. 4 - The figure shows a parabolic segment, that is. a...Ch. 4 - Given the point (a, b) in the first quadrant, find...Ch. 4 - The figure shows a region consisting of all points...Ch. 4 - Evaluate limn(1nn+1+1nn+2++1nn+n).Ch. 4 - For any number c, we let fc(x) be the smaller of...

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