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UFCalcSet17 - by x y 3 = y(a Find all points on the curve...

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Exercises UF Calculus Set 17 1. Suppose a ladder of length 25 feet rests against a wall, and we pull the base of the ladder horizontally away from the wall. For a given distance x that we pull the ladder horizontally, let g ( x ) be the height of the ladder on the wall, under the assumption that the other end of the ladder always contacts the wall. We know from Pythagorean theorem that for every x : x 2 + [ g ( x )] 2 = 25 2 (a) Write a formula for g 0 ( x ) in terms of x and g ( x ) . (b) At what rate is the height changing when x = 15 ? when the ladder is 5 feet from the ground ( g ( x ) = 5 )? (c) What do these rates measure? (Note the units.) 2. Find the derivative of the function y defined implicity in terms of x . (a) x + y + 2 x 2 + 3 y 3 = 2 (b) 3 xy 3 + 2 x 3 y = 1 (c) x y + 3 y = 2 (d) y = cos( x + y ) (e) y = xy + 1 (f) y = e x +2 y 3. Suppose y is implicitly defined as a function of x by the equations below. Show the point ( 1 , 1 ) is on the curve, and find the slope of the tangent line at that point. (a) 2 x y - y 2 = x (b) 1 + ye y = x + e x 4. Suppose y is implicitly defined as a function of
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Unformatted text preview: by x + y 3 = y . (a) Find all points on the curve with x-coordinate . (b) Find the slope of the curve at each point in part (a). (c) How many horizontal tangent lines does the curve have? (d) Find the points on the curve where the slope becomes vertical (vertical tangent lines). (e) Try to sketch the curve using the information above. Then justify your answer by consid-ering that the curve would correspond to the inverse function of portions of y = x 3-x . 5. Suppose y is implicitly defined as a function of x by 2 x 2-y 2 = 1 . Find the derivative and second derivative with respect to x in terms of x,y . 6. Find the derivative of each function below. (a) f ( x ) = arctan( e x ) (b) f ( x ) = arcsin( √ x ) (c) f ( x ) = arcsin( √ 1-x 2 ) (d) f ( x ) = (1 + x 2 ) p arctan( x ) ——————————————————————-...
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