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V63_0121_0007_2008F_Midterm

V63_0121_0007_2008F_Midterm - f ′ x(b(5 points Compute...

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Name: Midterm Exam Calculus I 2:00 - 3:15 pm, October 15, 2008 This examination booklet contains 6 problems on 7 sheets of paper including the front cover. Do all of your work in this booklet and show all your computations. This is a closed book exam. No calculators are allowed. 1 2 3 4 5 6 Total
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1. Compute the following limits (6 points each). (a) lim x →− 3 x 2 - 9 x 2 + 2 x - 3 (b) lim x →∞ 5 x 4 - 2 x + 1 - 2 x 4 + 4 x 3 + 3 (c) lim x 0 x 2 tan 2 (2 x )
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2. (15 points) Let f ( x ) = - x + 1 if x < 0 5 - x if 0 x < 3 ( x - 3) 2 + 2 if x > 3 (a) Compute the following limits lim x 0 f ( x ) , lim x 0+ f ( x ) , lim x 0 f ( x ) lim x 3 f ( x ) , lim x 3+ f ( x ) , lim x 3 f ( x ) (b) Where is the function discontinuous? Explain why.
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3. (a) (10 points) Let f ( x ) = x . Use the definition of derivative to compute
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Unformatted text preview: f ′ ( x ). (b) (5 points) Compute the equation oF the line tangent to f ( x ) at x = 4. 4. Calculate y ′ ( x ) (7 points each): (a) y = 2( x-1) 3 + √ 5 x-1 (b) y = x 2 cos( x + 2) (c) y = x + 1 1-x (d) y = 1 sin( x-cos( x )) 5. (12 points) Use implicit diferentiation to calculate y ′ ( x ): x tan(3 y ) = x 2 y + xy 2-1 6. (a) Use intermediate value theorem to show that f ( x ) = 2 x 3-4 x 2-5 x + 4 = 0 has a root in the interval [0 , 2]. (6 points) (b) Show that f ( x ) = 0 has another root. (6 points)...
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