# Exam 1 - 1 7 x a Write down the speciﬁc limit which is...

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MAT 2010 EXAM 1, February 11, 2011 Write your solutions in a bluebook. Show all your work. 1. (8 points.) Let y = f ( t ) = t 2 + 2 t . Find the Average Rate of Change of y with respect to t as t varies from t = 4 to t = 6. 2. (24 points.) Evaluate the following limits. (Write + or -∞ if appropriate. Show your reasoning, using the word “almost” when appropriate.) a) lim x 3 x 2 - x - 6 x - 3 b) lim x 0 2 cos( x )+6 e x + 4+ x c) lim x 0 - x 2 + x | x | 3. (8 points) Evaluate the following limit: (Write + or -∞ if appropriate. Show your reasoning, using the word “almost” when appropriate.) lim x →-∞ x 6 - 5 x 3 - x 6 4. (8 points) (a) The Intermediate Value Theorem implies that the equation 6 - x - x 5 = 0 has a solution on which of the following intervals: [ - 1 , 0], [1 , 0], or [1 , 2]? (Justify your choice(s).) (b) Find an interval of length 1/2 on which the equation has a solution. 5. (10 points.) Let f ( x ) =
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Unformatted text preview: 1 7+ x . a) Write down the speciﬁc limit which is the deﬁnition of f ( a ). b) Use the deﬁnition of derivative that you gave in part (a) to ﬁnd f ( a ). 6. (8 points.) Refer to Figure 6. Arrange the following quantities in numerical order (from negative to positive): , f ( a ) , f ( b ) , f ( c ) . 7. (10 points.) Refer to Figure 7. Depicted in Figure 7 is the graph of a function. Sketch as best you can the graph of its derivative, assuming that the derivative is continuous on the open intervals where it is deﬁned. 8. (24 points.) Find the derivatives for the following functions of x . Do not simplify: a) f ( x ) = 1 x 2 + x 3 e 4 x b) g ( x ) = sin(3 x )+ x 3 cos(5 x )+tan(2 x ) c) h ( x ) = ( x 2 + e 3 x + 2) 20 ....
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