UFCalcSet13 - ( 2 , 4 ) passes through the point ( 0 , 8 )...

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Exercises UF Calculus Set 13 1. Find the derivative of the given function. (a) f ( x ) = e x (b) f ( x ) = x e (c) f ( x ) = e e (d) f ( x ) = ex (e) f ( x ) = ex 2. Find the derivative of the given function. (a) f ( x ) = 1 + x 10 (b) f ( x ) = x 2 + π 2 (c) f ( x ) = x 3 - 2 x 2 + 4 x (d) f ( x ) = x 5 / 3 - x - 1 / 3 (e) f ( x ) = ( x 2 + 2)(2 x - 4) (f) f ( x ) = 2 + x x 2 3. Find an equation for the tangent line to the curve at the given point(s). Sketch the curve and the tangent line(s). (a) y = 1 + x x at ( 1 , 2 ) . (b) y = x 2 - 3 x + 2 at the intercepts ( 1 , 0 ) , ( 2 , 0 ) , and ( 0 , 2 ) . 4. Find the third derivative of y = x 2 - 2 x . 5. The height h (in feet) of an object shot into the air from a tall building is given by the function h ( t ) = 800 + 80 t - 16 t 2 , where t is the time elapsed in seconds. (a) Write a formula for the velocity of the object as a function of time t . (b) What should be the velocity of the object when it reaches its highest point? when does this occur? (c) What is the maximum height reached by the object? (d) Write a formula for the acceleration of the object as a function of time t . 6. Find a parabola, passing through the origin, whose tangent line at
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Unformatted text preview: ( 2 , 4 ) passes through the point ( 0 , 8 ) . Sketch the curve and tangent line. 7. Examine the function f ( x ) = x + 1 x . (a) Find the points on the curve at which the tangent lines pass through the point ( 1 , 1 ) . (b) Prove that no tangent line to the curve passes through the origin. (c) Prove that there is one tangent line passing though the point ( A,A ) , A 6 = 0 , and it occurs at the point with x-coordinate A 2 . (d) Sketch the curve with a graphing device and illustrate the ideas above. 8. Write a formula for the derivative of each piecewise-dened function. Sketch each curve. (a) f ( x ) = x x x 2 x < (b) f ( x ) = e x x x + 1 x < 9. Write a formula for the n th derivative of the functions: (a) f ( x ) = 1 x (b) f ( x ) = x -...
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UFCalcSet13 - ( 2 , 4 ) passes through the point ( 0 , 8 )...

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