30 Prove that d dx tan x sec 2 x and d dx sec x sec x tan x using well know

# 30 prove that d dx tan x sec 2 x and d dx sec x sec x

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30. Prove that d dx [tan( x )] = sec 2 ( x ) and d dx [sec( x )] = sec( x ) tan( x ) using well-know trigonometric iden- tities, the quotient rule, and the derivatives of sine and cosine. 31. Prove that d dx [arcsin( x )] = 1 1 - x 2 using implicit differentiation. 32. Prove that d dx [arctan( x )] = 1 x 2 + 1 using implicit differentiation. 33. Use logarithmic differentiation to prove that d dx [ b x ] = b x ln( b ). 34. Prove that d dx [ln( x )] = 1 x for x > 0 using implicit differentiation and the fact that d dx [ e x ] = e x . 35. Prove that d dx f - 1 ( x ) = 1 f 0 ( f - 1 ( x )) . Miscellaneous 36. If f (2) = 4 and f 0 (2) = 7 determine the derivative of f - 1 at 4. 37. If f ( x ) = 2 x - 1 3 x + 4 , determine d dx f - 1 ( x ) in two different ways. 38. If g ( d ) = ab 2 + 3 c 3 d + 5 b 2 c 2 d 2 , then what is g 00 ( d )? 39. If dy dx = 5 and dx dt = - 2 then what is dy dt ? 40. A ball is thrown into the air and its height h (in meters) after t seconds is given by the function h ( t ) = 10 + 20 t - 5 t 2 . When the ball reaches its maximum height, its velocity will be zero. (a) At what time will the ball reach its maximum height?
MAT 136: Calculus I Exam 2 Supplemental Problems (b) What is the maximum height of the ball? 41. Find the the values of x such that f 00 ( x ) = 0, where f ( x ) = x 5 20 - x 4 6 + x 3 6 + 5 x + 1 . 42. Find an equation of the tangent line to the graph of y = x 3 at x = 2? 43. Find an equation of the tangent line to the graph of y = 2 e x at x = 1? 44. Find an equation of the tangent line to the graph of x 2 / 3 + y 2 / 3 = 4 at the point ( - 3 3 , 1). 45. Suppose that h ( x ) = f ( x ) + g ( x ), where f ( x ) = 17 - x and the equation of the tangent line to g at x = 4 is y = 3 x + 10. Find h 0 (4). 46. If ln( x ) - y = 0, find dx dy . 47. Provide an example of a function f such that f is continuous at 0, but f 0 (0) does not exist. 48. Determine the value of the constant a so that f ( x ) = ( a 2 x - a, x 1 ax + 3 x 2 , x > 1 is continuous and differentiable. 49. The graphs of the functions f and h are given below. (a) Define g = fh . What is g 0 (2)? (b) Define k = f h . What is k 0 (2)? (c) Define m = f h . What is m 0 (2)? 50. Suppose f and g are differentiable functions such that f (3) = 2, f 0 (3) = 4, g (3) = 1, g 0 (3) = 3, and f 0 (1) = 5. (a) If h ( x ) = f ( x ) g ( x ), what is h 0 (3)? (b) If k ( x ) = f ( x ) g ( x ) , what is k 0 (3)? (c) If m ( x ) = f g ( x ), what is m 0 (3)?
MAT 136: Calculus I Exam 2 Supplemental Problems 51. Find the points on the curve x 2 + 2 y 2 = 6 where the slope of the tangent line is 1.

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