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119ncc08fallm1

# 119ncc08fallm1 - f x h = f x f h x 2 h xh 2 for all x h ∈...

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Last Name Name Student No Department: Section: Signature: : : : : : : : : : : Code Acad.Year Semester Date Time Duration Calculus and Analytical Geometry I. Midterm Math 119 2008-2009 Fall 4.11.2008 17:40 120 minutes 7 QUESTIONS ON 6 PAGES TOTAL 100 POINTS 1 2 3 4 5 6 7 1. (6+6+6=18 points) Evaluate the following limits, if they exist. Show your work. Do not use L’Hospital’s rule. (a) lim x 4 2 x + 1 - 3 x - 4 (b) lim x 0 x 3 sin( π/x ) (c) lim x 0 tan 2 x sin 3 x

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2. (10 points) Prove that lim x 2 4 x + 1 = 3 using the precise definition of a limit.
3. (15 points) Find the values of a and b which make the following function differentiable at each x R : f ( x ) = cos( πx ) , x 1 2 x 2 + ax + b, x > 1 2

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4. (6+6+6=18 points) Find the indicated derivatives. Do not simplify your answers in parts (a) and (b). (a) f ( x ) = x cos(sin( x 2 + 1)). Find f prime ( x ). (b) f ( x ) = x tan x 2 x + 1 . Find f prime ( x ). (c) f ( x ) = 1 3 x + 1 . Find f ( n ) ( x ) for each positive integer n .
5. (9 points) Suppose that f is a function that satisfies the following equation

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Unformatted text preview: f ( x + h ) = f ( x ) + f ( h ) + x 2 h + xh 2 for all x, h ∈ R . Suppose that lim h → f ( h ) h = 1. (a) ±ind f (0). (c) ±ind f p ( x ). 6. (15 points) Suppose that ABCD is a trapezoid with the sides AB and CD parallel. Suppose that | AB | is increasing at a rate of 1 cm/min , | CD | is decreasing at a rate of 3 cm/min and the height h of the trapezoid is increasing at a rate of 1 cm/min . Find the rate at which the area of the trapezoid is changing at an instant when | AB | = 5 cm , | CD | = 7 cm and h = 3 cm . 7. (15 points) Consider the curves x n + y n = 2 xy where n ≥ 3 is an integer. Show that all of these curves have the same tangent line through the point (1 , 1). Find an equation of this line....
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