hw5solutions

hw5solutions - is on the line it must be the case that y =...

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Math 2250 Written HW #5 Solutions 1. What is the rate of change of the volume of a ball (remember that the volume of a ball of radius r is given by 4 3 πr 3 ) with respect to the radius when the radius is r = 2? Answer: Think of the volume V of the ball as a function of r . Specifically, V ( r ) = 4 3 πr 3 . Therefore, the rate of change of the volume of the ball with respect to the radius when r = 2 is given by V 0 (2). Using the definition of the derivative, V 0 (2) = lim h 0 V (2 + h ) - V (2) h = lim h 0 ± 4 3 π (2 + h ) 3 ² - ± 4 3 π (2) 3 ² h = lim h 0 4 3 π ( 8 + 12 h + 6 h 2 + h 3 ) - 4 3 π · 8 h = lim h 0 16 πh + 8 πh 2 + 4 3 πh 3 h = lim h 0 ³ 16 π + 8 πh + 4 3 πh 2 ´ = 16 π. Hence, the desired rate of change is 16 π . 2. Show that the line y = mx + b is its own tangent line at any point ( x 0 ,y 0 ) on the line. Answer: Let f ( x ) = mx + b . Pick a point ( x 0 ,y 0 ) on the line y = mx + b . Since the point
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Unformatted text preview: is on the line, it must be the case that y = mx + b . Now, the slope of the tangent line to y = mx + b at the point ( x ,y ) is given by f ( x ). By definition, f ( x ) = lim h → f ( x + h )-f ( x ) h = lim h → [ m ( x + h ) + b ]-[ mx + b ] h = lim h → ( mx + mh + b )-( mx + b ) h = lim h → mh h = lim h → m = m. Therefore, by the point-slope formula, the tangent line is given by y-y = m ( x-x ) . 1 Using the fact that y = mx + b , this becomes y-mx-b = mx-mx . Adding mx + b to both sides tells us that the tangent line is y = mx + b, as desired. 2...
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hw5solutions - is on the line it must be the case that y =...

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