Explanation 1) Concept: The curvature of the curve given by the function f x is k x = f ' ' x 1 + f ' x 2 3 2 2) Given: y = e x 3) Calculation: The given equation is y = e x Consider f x = e x Differentiate with respect to x f ' x = e x Again, differentiate f ' ' x = e x Thus, the curvature of the curve f x is k x = f ' ' x 1 + f ' x 2 3 2 = e x 1 + e x 2 3 2 = e x 1 + e 2 x 3 2 To find maximum curvature by using first derivative test Differentiate k x k ' x = 1 + e 2 x 3 2 e x - e x 3 2 1 + e 2 x 3 2 - 1 e 2 x 2 1 + e 2 x 3 k ' x = 1 + e 2 x 3 2 e x - 3 e 3 x 1 + e 2 x 1 2 1 + e 2 x 3 k ' x = 1 + e 2 x 1 2 e x 1 + e 2 x - 3 e 2 x 1 + e 2 x 3 k ' x = e x 1 - 2 e 2 x 1 + e 2 x 5 2 So, to find the critical points of k x k ' x = 0 e x 1 - 2 e 2 x 1 + e 2 x 5 2 = 0 1 - 2 e 2 x = 0 e 2 x = 1 2 x = - 1 2 ln ⁡ 2 Check increasing and decreasing of k x for - ∞ , - 1 2 ln ⁡ 2 , - 1 2 ln ⁡ 2 , ∞ Take - 2 ∈ - ∞ , - 1 2 ln ⁡ 2 k ' - 2 = e - 2 1 - 2 e 2 - 2 1 + e 2 - 2 5 2 = 0

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ISBN: 9781305525924

Author: James Stewart

Publisher: Cengage Learning

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Chapter 13.3, Problem 31E

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The point which has maximum curvature of given function and what happens to the curvature as x→∞

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Chapter 13.3, Problem 30E

Chapter 13.3, Problem 32E

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Chapter 13 Solutions

Bundle: Calculus, 8th + Enhanced WebAssign - Start Smart Guide for Students + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Multi-Term

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Find the limit. limt1(t2tt1i+t+8j+sintlntk)Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Prob. 14E Ch. 13.1 - Prob. 15E Ch. 13.1 - Prob. 16E Ch. 13.1 - Prob. 17E Ch. 13.1 - Prob. 18E Ch. 13.1 - Prob. 19E Ch. 13.1 - Prob. 20E Ch. 13.1 - Prob. 21E 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 - Prob. 25E Ch. 13.1 - Prob. 26E Ch. 13.1 - Prob. 27E Ch. 13.1 - Prob. 28E Ch. 13.1 - Prob. 29E Ch. 13.1 - Prob. 30E Ch. 13.1 - Prob. 31E Ch. 13.1 - Prob. 32E Ch. 13.1 - Prob. 33E Ch. 13.1 - Prob. 34E Ch. 13.1 - Prob. 35E Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Prob. 40E Ch. 13.1 - Show that the curve with parametric equations...Ch. 13.1 - Prob. 42E Ch. 13.1 - Prob. 43E Ch. 13.1 - Prob. 44E Ch. 13.1 - Prob. 45E Ch. 13.1 - Prob. 46E Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Prob. 50E Ch. 13.1 - a Graph the curve with parametric equations...Ch. 13.1 - Prob. 52E Ch. 13.1 - Prob. 53E Ch. 13.1 - Prob. 54E Ch. 13.2 - Prob. 1E Ch. 13.2 - Prob. 2E Ch. 13.2 - Prob. 3E Ch. 13.2 - Prob. 4E Ch. 13.2 - Prob. 5E Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 7E Ch. 13.2 - Prob. 8E Ch. 13.2 - Prob. 9E Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Prob. 13E Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Prob. 18E Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - Prob. 21E Ch. 13.2 - Prob. 22E Ch. 13.2 - Prob. 23E Ch. 13.2 - Prob. 24E Ch. 13.2 - Prob. 25E Ch. 13.2 - Prob. 26E Ch. 13.2 - Prob. 27E Ch. 13.2 - Prob. 28E Ch. 13.2 - Prob. 29E Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Prob. 31E Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - Prob. 34E Ch. 13.2 - Evaluate the integral. 02(tit3j+3t5k)dt Ch. 13.2 - Prob. 36E Ch. 13.2 - Prob. 37E Ch. 13.2 - Prob. 38E Ch. 13.2 - Evaluate the integral. (sec2ti+t(t2+1)3j+t2lntk)dt Ch. 13.2 - Prob. 40E Ch. 13.2 - Prob. 41E Ch. 13.2 - Prob. 42E Ch. 13.2 - Prob. 43E Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prove Formula 5 of Theorem 3.Ch. 13.2 - Prob. 46E Ch. 13.2 - Prob. 47E Ch. 13.2 - If u and v are the vector functions in Exercise...Ch. 13.2 - Prob. 49E Ch. 13.2 - Prob. 50E Ch. 13.2 - If r(t)=acost+bsint, where a and b are constant...Ch. 13.2 - Prob. 52E Ch. 13.2 - Prob. 53E Ch. 13.2 - Find an expression for ddt[u(t)(v(t)w(t))].Ch. 13.2 - Prob. 55E Ch. 13.2 - Prob. 56E Ch. 13.2 - Prob. 57E Ch. 13.2 - Prob. 58E Ch. 13.3 - Find the length of the curve....Ch. 13.3 - Prob. 2E Ch. 13.3 - Prob. 3E Ch. 13.3 - Prob. 4E Ch. 13.3 - Find the length of the curve. r(t)=i+t2j+t3k,0t1 Ch. 13.3 - Prob. 6E Ch. 13.3 - Prob. 7E Ch. 13.3 - Find the length of the curve correct of four...Ch. 13.3 - Prob. 9E Ch. 13.3 - Graph the curve with parametric equations...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 - Prob. 15E 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 - Prob. 18E Ch. 13.3 - Prob. 19E Ch. 13.3 - Prob. 20E Ch. 13.3 - Use Theorem 10 to find the curvature. r(t)=t3j+t2k Ch. 13.3 - Use Theorem 10 to find the curvature....Ch. 13.3 - Prob. 23E Ch. 13.3 - Find the curvature of r(t)=t2,lnt,tlnt at the...Ch. 13.3 - Find the curvature of r(t)=t,t2,t3 at the point...Ch. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Use Formula 11 to find the curvature. y=x4 Ch. 13.3 - Prob. 28E Ch. 13.3 - Use Formula 11 to find the curvature. y=xex Ch. 13.3 - Prob. 30E Ch. 13.3 - Prob. 31E 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 - Prob. 34E Ch. 13.3 - Prob. 35E Ch. 13.3 - Prob. 36E Ch. 13.3 - Prob. 37E Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Prob. 40E Ch. 13.3 - Prob. 41E Ch. 13.3 - Prob. 42E Ch. 13.3 - Prob. 43E Ch. 13.3 - Prob. 44E Ch. 13.3 - Prob. 45E Ch. 13.3 - Prob. 46E Ch. 13.3 - Prob. 47E Ch. 13.3 - Prob. 48E 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 - Find equations of the osculating circles of the...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - Prob. 53E Ch. 13.3 - Is there a point on the curve in Exercise 53 where...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Prob. 56E Ch. 13.3 - Show that at every point on the curve...Ch. 13.3 - Prob. 58E Ch. 13.3 - Prob. 59E Ch. 13.3 - Prob. 60E Ch. 13.3 - a Show that dB/ds is perpendicular to B. b Show...Ch. 13.3 - Prob. 62E Ch. 13.3 - Use the Frenet-Serret formulas to prove each of...Ch. 13.3 - Show that the circular helix r(t)=acost,asint,bt,...Ch. 13.3 - Prob. 65E Ch. 13.3 - Prob. 66E Ch. 13.3 - Prob. 67E Ch. 13.3 - Prob. 68E Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - The figure shows the path of a particle that moves...Ch. 13.4 - Prob. 3E Ch. 13.4 - Prob. 4E Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 6E Ch. 13.4 - Prob. 7E Ch. 13.4 - Prob. 8E Ch. 13.4 - Prob. 9E Ch. 13.4 - Prob. 10E Ch. 13.4 - Prob. 11E Ch. 13.4 - Prob. 12E 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 - Prob. 15E Ch. 13.4 - Prob. 16E Ch. 13.4 - a Find the position vector of a particle that has...Ch. 13.4 - Prob. 18E Ch. 13.4 - The position function of a particle is given by...Ch. 13.4 - Prob. 20E 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 - Prob. 24E Ch. 13.4 - Prob. 25E Ch. 13.4 - A projectile is fired 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 - Prob. 32E Ch. 13.4 - Water traveling along a straight portion of a...Ch. 13.4 - Prob. 34E Ch. 13.4 - Prob. 35E Ch. 13.4 - Prob. 36E Ch. 13.4 - Prob. 37E Ch. 13.4 - Prob. 38E Ch. 13.4 - Prob. 39E Ch. 13.4 - Prob. 40E Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Prob. 42E Ch. 13.4 - The magnitude of the acceleration vector a is 10...Ch. 13.4 - Prob. 44E Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - Prob. 46E Ch. 13.R - Prob. 1CC Ch. 13.R - Prob. 2CC Ch. 13.R - Prob. 3CC Ch. 13.R - Prob. 4CC Ch. 13.R - Prob. 5CC Ch. 13.R - Prob. 6CC Ch. 13.R - Prob. 7CC Ch. 13.R - Prob. 8CC Ch. 13.R - Prob. 9CC Ch. 13.R - Prob. 1TFQ Ch. 13.R - Prob. 2TFQ Ch. 13.R - Prob. 3TFQ Ch. 13.R - Prob. 4TFQ Ch. 13.R - Prob. 5TFQ Ch. 13.R - Prob. 6TFQ Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Prob. 8TFQ Ch. 13.R - Prob. 9TFQ Ch. 13.R - Prob. 10TFQ Ch. 13.R - Prob. 11TFQ Ch. 13.R - Prob. 12TFQ Ch. 13.R - Prob. 13TFQ Ch. 13.R - Prob. 14TFQ Ch. 13.R - Prob. 1E Ch. 13.R - Prob. 2E Ch. 13.R - Prob. 3E Ch. 13.R - Prob. 4E Ch. 13.R - Prob. 5E Ch. 13.R - Prob. 6E Ch. 13.R - Prob. 7E Ch. 13.R - Prob. 8E Ch. 13.R - Prob. 9E Ch. 13.R - Prob. 10E Ch. 13.R - For the curve given by r(t)=sin3t,cos3t,sin2t,...Ch. 13.R - Find the curvature of the ellipse x=3cost,y=4sint...Ch. 13.R - Find the curvature of the curve y=x4 at the point...Ch. 13.R - Find an equation of the osculating circle of the...Ch. 13.R - Prob. 15E Ch. 13.R - The figure shows the curve C traced by a particle...Ch. 13.R - A particle moves with position function...Ch. 13.R - Prob. 18E Ch. 13.R - A particle starts at the origin with initial...Ch. 13.R - Prob. 20E Ch. 13.R - A projectile is launched with an initial speed of...Ch. 13.R - Prob. 22E Ch. 13.R - Prob. 23E Ch. 13.R - In designing transfer curves to connect sections...Ch. 13.P - A particle P moves with constant angular speed ...Ch. 13.P - A circular curve of radius R on a highway is...Ch. 13.P - A projectile is fired from the origin with angle...Ch. 13.P - a A projectile is fired from the origin down an...Ch. 13.P - A ball rolls off a table with a speed of 2 ft/s....Ch. 13.P - Prob. 6P Ch. 13.P - If a projectile is fired with angle of elevation ...Ch. 13.P - Prob. 8P Ch. 13.P - Prob. 9P

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The point which has maximum curvature of given function and what happens to the curvature as x → ∞

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Chapter 13 Solutions

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The point which has maximum curvature of given function and what happens to the curvature as x→∞