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Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550
Chapter 13, Problem 7RE
Textbook Problem
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Use Simpson’s Rule with n = 6 to estimate the length of the arc of the curve with equations x = t2, y = t3, z = t4, 0 ≤ t ≤ 3.

To determine

To estimate: The length of the arc of the curve with equations x=t2,y=t3,z=t4,0t3 .

Explanation of Solution

Formula used:

Write the expression to find the length of the curve r(t),t1tt2 .

L=t1t2|r(t)|dt (1)

Write the required differentiation formula to find the tangent vector r(t) .

ddttn=ntn1

Write the Simpson’s rule to find the length of the curve with the function f(t) and step length Δt .

t1t2f(t)dt=Δt3[f(0)+4f(Δt)+2f(2Δt)+4f(3Δt)+...+f(nΔt)] (2)

Here,

Δt is the step length and

n is the number, which considered based on the limits of scalar parameter t .

Write the expression to find the step length.

Δt=t2t1n (3)

Write the equation of the curve as follows.

x=t2,y=t3,z=t4,0t3

Write the vector equation from the equation of the curve as follows.

r(t)=t2,t3,t4

Calculation of derivative of the vector function r(t) [r(t)] :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=t2,t3,t4 as follows.

ddt[r(t)]=ddt(t2),ddt(t3),ddt(t4)

Rewrite and compute the expression as follows.

r(t)=2t21,3t31,4t41=2t,3t2,4t3

Find |r(t)| from the vector r(t)=2t,3t2,4t3 as follows.

|r(t)|=|2t,3t2,4t3|=(2t)2+(3t2)2+(4t3)2=4t2+9t4+16t6

Calculation of length of the curve (L) :

Substitute 4t2+9t4+16t6 for |r(t)| , 0 for t1 , and 3 for t2 in equation (1),

L=03(4t2+9t4+16t6)dt

Simplify and compute the expression by using Simpson’s rule as follows.

Consider |r(t)|=4t2+9t4+16t6 as function f(t) .

f(t)=4t2+9t4+16t6 (4)

As the limits of the scalar parameter is 0 to 3, consider the number n as 6 for simple and accurate calculations.

Calculation of step length Δt :

Substitute 0 for t1 , 3 for t2 , and 6 for n in equation (3),

Δt=306=12

Rewrite the Simpson’s rule by substituting L for t1t2f(t)dt , 12 for Δt , and 6 for n in equation (2) as follows

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Chapter 13 Solutions

Calculus: Early Transcendentals
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Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Draw the projections of the curve on the three...Ch. 13.1 - Draw the projections of the curve on the three...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Find three different surfaces that contain the...Ch. 13.1 - Find three different surfaces that contain the...Ch. 13.1 - At what points does the curve r(t) = t i + (2t ...Ch. 13.1 - At what points does the helix r(t) = sin t, cos t,...Ch. 13.1 - Graph the curve with parametric equations x = sin...Ch. 13.1 - Graph the curve with parametric equations x = (1 +...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Two particles travel along the space curves r1 (t)...Ch. 13.1 - Suppose u and v are vector functions that possess...Ch. 13.2 - The figure shows a curve C given by a vector...Ch. 13.2 - (a) Make a large sketch of the curve described by...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - Find the derivative of the vector function. 9....Ch. 13.2 - Find the derivative of the vector function. 10....Ch. 13.2 - Find the derivative of the vector function. 11....Ch. 13.2 - Find the derivative of the vector function. 12....Ch. 13.2 - Find the derivative of the vector function. 13....Ch. 13.2 - Find the derivative of the vector function. 14....Ch. 13.2 - Find the derivative of the vector function. 15....Ch. 13.2 - Find the derivative of the vector function. 16....Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - If r(t) = t, t2, t3, find r'(t), T( 1), r"(t). and...Ch. 13.2 - If r(t) = e2t, e2t, te2t, find T(0), r"(0), and...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find a vector equation for the tangent line to the...Ch. 13.2 - Find the point on the curve r(t) = 2 cos t, 2 sin...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - (a) Find the point of intersection of the tangent...Ch. 13.2 - The curves r1(t) = t, t2, t3 and r2(t) = sin t,...Ch. 13.2 - At what point do the curves r1(t) = t, 1 t, 3 +...Ch. 13.2 - Evaluate the integral. 35. 02(ti-t3j+3t5k)dtCh. 13.2 - Evaluate the integral. 36. 14(2t3/2i+(t+1)tk)dtCh. 13.2 - Evaluate the integral. 37....Ch. 13.2 - Evaluate the integral. 38....Ch. 13.2 - Evaluate the integral. 39....Ch. 13.2 - Evaluate the integral. 40. (te2ti+t1-tj+11-t2k)dtCh. 13.2 - Find r(t) if r'(t) = 2t i + 3t2 j + t k and r(1) =...Ch. 13.2 - Find r(t) if r'(t) = t i + et j + tet k and r(0) =...Ch. 13.2 - Prove Formula 1 of Theorem 3.Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prove Formula 5 of Theorem 3.Ch. 13.2 - Prove Formula 6 of Theorem 3.Ch. 13.2 - If u(t) = sin t, cos t, t) and v(t) = t, cos t,...Ch. 13.2 - If u and v are the vector functions in Exercise...Ch. 13.2 - Find f'(2), where f(t) = u(t) v(t), u(2) = 1, 2,...Ch. 13.2 - If r(t) = u(t) v(t), where u and v are the vector...Ch. 13.2 - If r(t) = a cos t + b sin t, where a and b are...Ch. 13.2 - If r is the vector function in Exercise 51, show...Ch. 13.2 - Show that if r is a vector function such that r''...Ch. 13.2 - Find an expression for ddt[u(t)(v(t)w(t))].Ch. 13.2 - If r(t) 0, show that ddtr(t)=1r(t)r(t)r(t)....Ch. 13.2 - If a curve has the property that the position...Ch. 13.2 - If u(t) = r(t)[r'(t) r''(t)], show that u(t) =...Ch. 13.2 - Show that the tangent vector to a curve defined by...Ch. 13.3 - Find the length of the curve. 1. r(t) =t, 3 cos t,...Ch. 13.3 - Find the length of the curve. 2. r(t)=2t,t2,13t3,...Ch. 13.3 - Find the length of the curve. 3. r(t)=2ti+etj+etk,...Ch. 13.3 - Find the length of the curve. 4. r(t) =cos t i +...Ch. 13.3 - Find the length of the curve. 5. r(t) = i + t2 j +...Ch. 13.3 - Find the length of the curve. 6. r(t) = t2 i + 9t...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Graph the curve with parametric equations x = sin...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Find, correct to four decimal places, the length...Ch. 13.3 - (a) Find the arc length function for the curve...Ch. 13.3 - (a) Find the arc length function for the curve...Ch. 13.3 - Suppose you start at the point (0, 0. 3) and move...Ch. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - Use Theorem 10 to find the curvature. 21. r(t) =...Ch. 13.3 - Use Theorem 10 to find the curvature. 22. r(t) = t...Ch. 13.3 - Use Theorem 10 to find the curvature. 23....Ch. 13.3 - Find the curvature of r(t) = t2, ln t, t ln t at...Ch. 13.3 - Find the curvature of r(t) = t, t2, t3 at the...Ch. 13.3 - Graph the curve with parametric equations x = cos...Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - To find: The curvature of y=tanx using Formula 11....Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - (a) Is the curvature of the curve C shown in the...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve y =...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve y =...Ch. 13.3 - Use Theorem 10 to show that the curvature of a...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Consider the curvature at x = 0 for each member of...Ch. 13.3 - Find the vectors T, N, and B at the given point....Ch. 13.3 - Find the vectors T, N, and B at the given point....Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - At what point on the curve x = t3, y = 3t, z = t4...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Show that the osculating plane at every point on...Ch. 13.3 - The rectifying plane of a curve at a point is the...Ch. 13.3 - Show that the curvature is related to the tangent...Ch. 13.3 - Show that the curvature of a plane curve is =...Ch. 13.3 - To deduce: the Formula dNds=KT+B. Solution: From...Ch. 13.3 - Use ihe Frenet-Serret formulas to prove each of...Ch. 13.3 - Show that the circular helix r(t) = a cos t, a sin...Ch. 13.3 - Use the formula in Exercise 63(d) to find the...Ch. 13.3 - Find the curvature and torsion of the curve x =...Ch. 13.3 - The DNA molecule has the shape of a double helix...Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - The position function of a particle is given by...Ch. 13.4 - What force is required so that a particle of mass...Ch. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Rework Exercise 23 if the projectile is fired from...Ch. 13.4 - A ball is thrown at an angle of 45 to the ground....Ch. 13.4 - A projectile is tired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - A batter hits a baseball 3 ft above the ground...Ch. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Show that a projectile reaches three-quarters of...Ch. 13.4 - A ball is thrown eastward into the air from the...Ch. 13.4 - A ball with mass 0.8 kg is thrown southward into...Ch. 13.4 - Another reasonable model for the water speed of...Ch. 13.4 - A particle has position function r(t). If r(t) = c...Ch. 13.4 - (a) If a particle moves along a straight line,...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - If a particle with mass m moves with position...Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - A rocket burning its onboard fuel while moving...Ch. 13 - What is a vector function? How do you find its...Ch. 13 - What is the connection between vector functions...Ch. 13 - How do you find the tangent vector to a smooth...Ch. 13 - If u and v are differentiable vector functions, c...Ch. 13 - How do you find the length of a space curve given...Ch. 13 - (a) What is the definition of curvature? (b) Write...Ch. 13 - (a) Write formulas for the unit normal and...Ch. 13 - (a) How do you find the velocity, speed, and...Ch. 13 - State Keplers Laws.Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Determine whether the statement is true or false....Ch. 13 - (a) Sketch the curve with vector function r(t) = t...Ch. 13 - Let r(t) = 2-t, (et 1)/t, ln(t + 1). (a) Find the...Ch. 13 - Find a vector function that represents the curve...Ch. 13 - Find parametric equations for the tangent line to...Ch. 13 - If r(t) = t2 i + t cos t j + sin t k, evaluate...Ch. 13 - Let C be the curve with equations x = 2 t3 y = 2t...Ch. 13 - Use Simpsons Rule with n = 6 to estimate the...Ch. 13 - Find the length of the curve r(t) = 2t3/2, cos 2t,...Ch. 13 - The helix r1(t) = cos t i + sin t j + t k...Ch. 13 - Reparametrize the curve r(t) = et i + et sin t j +...Ch. 13 - For the curve given by r(t) = sin3 t, cos3 t, sin2...Ch. 13 - Find the curvature of the ellipse x = 3 cos t, y =...Ch. 13 - Find the curvature of the curve y = x4 at the...Ch. 13 - Find an equation of the osculating circle of the...Ch. 13 - Find an equation of the osculating plane of the...Ch. 13 - The figure shows the curve C traced by a particle...Ch. 13 - A particle moves with position function r(t) = t...Ch. 13 - Find the velocity, speed, and acceleration of a...Ch. 13 - A particle starts at the origin with initial...Ch. 13 - An athlete throws a shot at an angle of 45 to the...Ch. 13 - A projectile is launched with an initial speed of...Ch. 13 - Find the tangential and normal components of the...Ch. 13 - A disk of radius 1 is rotating in the...Ch. 13 - PROBLEM PLUS FIGURE FOR PROBLEM 1 1. A particle P...Ch. 13 - A circular curve of radius R on a highway is...Ch. 13 - A projectile is fired from the origin with angle...Ch. 13 - (a) A projectile it fired from the origin down an...Ch. 13 - A ball rolls off a table with a speed of 2 ft/s....Ch. 13 - Find the curvature of the curve with parametric...Ch. 13 - If a projectile is fired with angle of elevation ...Ch. 13 - A cable has radius r and length L and is wound...Ch. 13 - Show that the curve with vector equation r(t) =...

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