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7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 12.5, Problem 44E

Textbook Problem

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Finding Curvature in Rectangular Coordinates In Exercises 41-48, find the curvature and radius of curvature of the plane curve at the given value of r.

To determine

**To calculate:** The curvature and radius of the curvature of the plane curve in the rectangular coordinates *x*.

**Given:**

The rectangular coordinate is

**Formula Used:**

The curvature *K* at the point *r* is

Chain rule: derivative of

Quotient rule: derivative of

**Calculation:**

Consider function

Differentiating the function with respect to *x* by applying chain rule,

Now, again differentiate with respect to *x* by using quotient rule,

Calculus: Early Transcendental Functions

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What...Ch. 12.5 - Curvature Given a twice-differentiable function...Ch. 12.5 - Investigation Consider the function f(x)=x4x2. (a)...Ch. 12.5 - Motion of a Particle A particle moves along the...Ch. 12.5 - Investigation Find all a and b such that the two...Ch. 12.5 - HOW DO YOU SEE IT? Using the graph of the ellipse,...Ch. 12.5 - Sphere and Paraboloid A sphere of radius 4 is...Ch. 12.5 - The smaller the curvature of a bend in a road, the...Ch. 12.5 - Center of Curvature Let C be a curve given by...Ch. 12.5 - Center of Curvature Use the result of Exercise 67...Ch. 12.5 - Curvature A curve C is given by the polar equation...Ch. 12.5 - Curvature Use the result of Exercise 69 to find...Ch. 12.5 - Curvature Given the polar curve r=ea,a0, use the...Ch. 12.5 - Curvature at the Pole Show. that the formula for...Ch. 12.5 - Curvature at the Pole In Exercises 73 and 74, use...Ch. 12.5 - Curvature at the Pole In Exercises 73 and 74, use...Ch. 12.5 - Proof For a smooth curve given by the parametric...Ch. 12.5 - Horizontal Asymptotes Use the result of Exercise...Ch. 12.5 - Curvature of a Cycloid Use the result of Exercise...Ch. 12.5 - Tangential and Normal Components of Acceleration...Ch. 12.5 - Frictional Force A 5500-pound vehicle is driven at...Ch. 12.5 - Frictional Force A 6400-pound vehicle is driven at...Ch. 12.5 - Curvature Verify that the curvature at any point...Ch. 12.5 - Formulas for Curvature Use the definition of...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Show that rGMLrr=e is a constant vector.Ch. 12.5 - Prove Keplers First Law: Each planet moves in an...Ch. 12.5 - Assume that the elliptical orbit r=ed1+ecos is in...Ch. 12.5 - Prove Keplers Second Law: Each ray from the sun to...Ch. 12.5 - Prove Keplers Third Law: The square of the period...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - 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Finding the Unit Tangent Vector In Exercises 43...Ch. 12 - Finding a Tangent Line In Exercises 45 and 46,...Ch. 12 - Finding a Tangent Line In Exercises 45 and 46,...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - r(t)=2ti3tj,[0,5] Finding the Arc Length of a...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 67 and 68, find the...Ch. 12 - Finding Curvature In Exercises 67 and 68, find the...Ch. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Frictional Force A 7200-pound vehicle is driven at...Ch. 12 - Cornu Spiral The cornu spiral is given by...Ch. 12 - Radius of Curvature Let T be the tangent line at...Ch. 12 - Projectile Motion A bomber is flying horizontally...Ch. 12 - Projectile Motion Repeat Exercise 3 for the case...Ch. 12 - Cycloid Consider one arch of the cycloid as shown...Ch. 12 - 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