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02_Midterm1-Sample

# 02_Midterm1-Sample - &nd the derivative of f x(b(3...

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University of California, Riverside Department of Mathematics Midterm 1 Mathematics 9A - First Year of Calculus Sample 2 Instructions: This exam has a total of 33 points. You have 50 minutes. You must show all your work to receive full credit You may use any result done in class. The points attached to each problem are indicated beside the problem. You are not allowed to use books, notes, or calculators. Answers should be written as p 2 as opposed to 1 : 4142135 ::: . 1. Find each of the following limits if it exists. If you think the limit does not exist provide a reason. (a) (3 points) lim x ! 4 4 °j x j p x ° 2 (b) (3 points) lim t ! 2 + 4 t 6 ° 3 t (c) (3 points) lim t !1 7 x 12 1+45 x ° 14 x 12 2. (6 points) Let f ( x ) = 7 ° 3 x and let L = 1 : Prove that L = lim x ! 2 f ( x ) using the de°nition of limits, i.e. For every given " > 0 , °nd ° > 0 such that: 0 < j x ° 2 j < ° ) j f ( x ) ° L j < " : 3. Let f ( x ) = 5 ° x 2 : (a) (3 points) Using the de°nition

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Unformatted text preview: , &nd the derivative of f ( x ) . (b) (3 points) Find the equation of the tangents line to the graph of f ( x ) at x = 2 . 4. Find the derivatives of the following functions: (a) (3 points) f ( t ) = 2 t 3 & t 3 1 (b) (3 points) y = 1 5 x & p 2 & 3 x 2 & & 2 5. Consider the following graph of a function y = f ( x ) : f ( x ) = 8 > > < > > : & x & 2 x < & 2 ( x + 2) 2 & 2 ± x < ( x & 1) 2 + 3 ± x < 2 x & 3 2 ± x-5-4-3-2-1 1 2 3 4 5-1 1 2 3 4 x y (a) (2 points) Find the following limits if they exists: i. lim x ! 2 + f ( x ) ii. lim x !& 2 f ( x ) (b) (2 points) List all x-values where f is discontinuous. Give reasons for your answers. (c) (2 points) List all x-values where f is continuous but not di/er-entiable. 2...
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