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### Fall 2009 Final with Solutions

Course: MATH 103, Fall 2012
School: UPenn
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Word Count: 473

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103 Final Math Exam Fall 2009 3 6 0 1. If 6 0 3 f ( x ) dx = 12 and f ( x ) dx = 42 , find the value of ( 2 f ( x ) 3) dx. a) 5 0 b) 51 c) 5 2 d) 56 e) 53 f) 54 g) 55 h) None of these 8 2. Find the value of the integral 3 0 1 x + 1 dx. 32 x a) 14 b) 6 c) 12 e) 18 f) 72 g) e2 3. Find the value of ( ln x ) e d) 20 5 2 h) 168 2 x dx. a) ln2 b) 1 ln 2 2 c) 1 2 d) 3...

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103 Final Math Exam Fall 2009 3 6 0 1. If 6 0 3 f ( x ) dx = 12 and f ( x ) dx = 42 , find the value of ( 2 f ( x ) 3) dx. a) 5 0 b) 51 c) 5 2 d) 56 e) 53 f) 54 g) 55 h) None of these 8 2. Find the value of the integral 3 0 1 x + 1 dx. 32 x a) 14 b) 6 c) 12 e) 18 f) 72 g) e2 3. Find the value of ( ln x ) e d) 20 5 2 h) 168 2 x dx. a) ln2 b) 1 ln 2 2 c) 1 2 d) 3 2 e) 1 f) 1 ln 2 g) 0 h) 7 3 ln 9 4. Find the value of e x /2 dx. 0 a) 4 b) 2 c) 6 d) 3 e) 16 f) 9 g) 8 h) 1 5. Find all critical numbers for the function f ( x ) = 3 9 x 2 . a) 0 b) -3 c) 0, -3 d) No critical numbers e) 3 f) 3, -3 g) 0, 3, -3 h) None of these x2 + 4 x + 5 if x < 2 1 6. At what value(s) of x is the function f ( x ) = if 2 x 2 discontinuous? 2x if x > 2 1+ x 2 a) -2 b) 0 c) -2,0, and 2 d) -2 and 0 e) 2 f) -2 and 2 g) 0 and 2 h) f is continuous everywhere ( ) 7. Find the interval on which the graph of f ( x ) = ln x 2 + 1 is concave upward. a) (-1,1) b) (-1,2) c) (-2,1) d) (-2,2) e) (-1,3) f) (-3,2) g) (-3,3) h) ( , ) 8. The curve y = x 3 + x 2 x has two horizontal tangents. Find the distance between these two horizontal lines. a) 11 9 b) 22 27 c) 32 27 d) 5 3 e) 14 9 f) 4 3 g) 13 9 h) 7 3 9. If f ( x ) = a) x , find f . tan x 4 2 2 b) e) 1 2 1 2 x ln ( 3 + 2e5 x ) 6x a) 5 b) 1 5 c) 1 6 f) 1 10 g) 2 10 e) 5 3 d) g) 2 f) 2 10. Evaluate the limit lim c) 1 h) . d) 3 5 h) 5 6 2 11. Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time. When the height is 5 in. and is increasing at 0.2 in./s. , the radius is 3 in. and is decreasing at 0.1 in./s. How fast is the volume changing at that instant? 12. A rectangle with base on the x -axis has its upper vertices on the curve y = 12 x 2 . Find the maximum area of such a rectangle. Be sure to prove that you have found the maximum area. x 13. The graph of f below consists of line segments and semicircles. Let g ( x ) = f ( t ) dt . 0 Answer the following questions. (a) g (14 ) (b) g (10 ) (c) g ( 6 ) (d) What is the absolute minimum value of g on the interval [0,14]? 14. Given the graph of y = f ( x ) , answer the questions that follow. (a) Find all values of x at which (Explain your answers for full credit) (i) f is increasing. (ii) f is decreasing. (iii) f ( x ) > 0. (iv) f has an inflection point. (v) f has a local maximum (b) Sketch a graph which could represent y = f ( x ) . Math 103 Fall 2009 Final Exam Answers 1. B 2. E 3. H 4. A 5. G 6. A 7. A 8. C 9. A 10. H 11. 6 5 12. 32 un.2 13. a) 4 + 5 2 b) 12 + 9 2 c) 2 d) 0 14. a) i) ( x1 , x3 ) ii) ( x0 , x1 ) ( x3 , x4 ) iii) ( x0 , x2 ) iv) x2 v) x3 b) x0 x1 x2 x3 x4
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