1. True / False. Every function f that is differentiable on the closed interval [ a , b ] is itself the derivative of some function g on the same closed interval [ a , b ] . Explain your reasoning if it is true or give a counterexample if it is false. 1) ______ ( A ) True ( B ) False ( C ) not enoughinformation of the above ( D) 2. True/False. If f ( x ) is an even function and g ( x ) is even function, then ( f + g )( x ) function. Explain your reasoning if it is true or give a counterexample if it is false. is an even 2) ______ ( A ) True ( B ) False ( C ) not enoughinformation 1 of the above ( D) 3. For which numbers ax +b cx +d satisfy f ( f ( x ) ) =x a , b , c , and d will the function? f ( x )= for all x ? 3) ______ of the above ( A ) a+b+ c+ d 0 ( B ) c+ d 0 ( C ) a2 +bc 0 ( D ) not enoughinformation ( E ) 2 4. Find the limit 2 lim x 0 sin 2 x 2 x 4) ______ of the above ( A ) 0 (B )1 (C) 2( D ) 4 ( E) 5. Fill in the blank. If lim f ( x ) =L , then ? Prove your answer is true. x a lim |f |( x )= x a of the above ( A ) L ( B )|L|(C ) f ( a ) ( D )|f |( a )( E ) 5) ______ 3 6. Find the limit lim x 2 x ( x +sin x ) 2 6) _____ of the above ( A ) 0 ( B ) 1/ 2 (C ) 1 ( D ) does not exist ( E ) 2 7. True/False. There exists a non-constant function f such that ( f ( x ) ) =x 2 . If your answer is true, then give all of the functions that satisfy this condition. If the answer is false, explain. 7) ______ of the above ( A ) True ( B ) False ( C ) not enough information ( D ) 4 8. True/False. If f is differentiable at c , then |f | is differentiable at reasoning if you answered true. Give a counterexample if you answered false. c . Explain your 8) ______ of the above ( A ) True ( B ) False ( C ) not enough information ( D ) 9. Find ' f ( x ) if f is a function defined as f ( x )= where { 1 , x 0 x 0, x=0 g ( x ) sin g ( 0 )=g' ( 0 )=0 . 9) ______ of the above ( A ) 0 ( B ) 1 ( C ) undefined ( D ) not enough information ( E ) 5 10. Find g' ( 0 ) if g ( x ) =xf ( x ) for some function f which is continuous at 0. 10) ______ of the above ( A ) 0 ( B ) g ( 0 ) ( C ) f ( 0 )( D ) x f ' ( 0 ) ( E ) 11. True/False. If f ( x ) g ( x ) =x , then it is possible that f and g are differentiable and f ( 0 )=g ( 0 )=0 . If you answered true, then explain your reasoning. If you answered false, then give a counterexample. 11) ______ of the above ( A ) True ( B ) False ( C ) not enough information ( D ) 6 12. Find the minimum value for the sum of a number and its reciprocal. 12) ______ of theabove ( A ) 0 ( B ) 1 ( C ) 2 ( D ) there is no minimum value ( E ) 7 13. Two hallways of width a and b meet at right angles. Find the greatest possible length of a ladder that can be carried horizontally around the corner. 13) _____ of the above 3/ 2 ( A ) a + b ( B ) a+ b ( C ) a2 /3 +b 2/3 ( D ) ( a2 /3 +b 2/ 3) ( E ) 2 2 8 14. Use the graph to determine whether the function has absolute extreme values on the interval ( a , b ) . Explain your reasoning. 14) ______ (A) Absolute minimum and absolute maximum (B) Absolute minimum only (C) Absolute maximum only (D) No absolute extrema (E) none of the above y a b 9 x 15. The acceleration, a= ds d2 s , initial velocity v = , and initial position s ( t ) 2 dt dt moving along a coordinate line are given beow. Find the particle's position at time of a particle t . So, a ( t )=e t ; v ( 0 ) =12 ; s ( 0 )=14 . 15) ______ ( A ) s ( t )=et +11 t +13 ( B ) s ( t )=e t +11 t ( C ) s ( t ) =e t +12t +14 ( D ) s ( t ) et +13 of the above (E ) 16. The graph of a function f is shown. Which graph is an antiderivative of f and why? 16) _______ D 10 C f B A of the above ( A ) Graph A ( B ) Graph B ( C ) Graph C ( D ) Graph D ( E ) 17. Evaluate the definite integral 2 2 ( x+ 1x ) dx 1 17) ______ of the above 37 29 15 5 ( A ) (B ) (C ) ( D) ( E) 6 6 2 6 11 18. Find x y' if 2 y= 2 t+ 4 dt 1 18) ______ of the above 2 1 1 ( A ) 2 x 2 x + 4 ( B ) 2 x + 4 ( C ) ( D ) ( 2 x 2 + 4 6 ) 6 ( E ) 3 2t +4 3 2 2 19. Raindrops increase in size as they fall and so their resistance to falling increases. Suppose a raindrop has an initial downward velocity of 10 m/sec and its downward acceleration is { a ( t )= 90.9 t ,if 0 t 10 0, if t>10 If the raindrop is initially 500 m above the ground, how long does it take to fall? 19) ______ ( A ) 9.8 s ( B ) 10 s ( C ) 11.8 s ( D ) 13.6 s ( E ) n one of the above 12 20. True/False. If ( a , b ) such that x f is integrable on [ a , b] , then there exists a number x in the open interval b f = f a x If your answer is true, then explain your reasoning. If you chose false, then give a counterexample. 20) ______ of the above ( A ) True ( B ) False ( C ) not enough information ( D ) 13 21. EXTRA CREDIT: True/False. If f and g are both integrable on [ a , b ] , then f o g is also integrable on [ a , b ] . If true, explain your reasoning. If false, give a counterexample. 21) ______ of the above ( A ) True ( B ) False ( C ) not enough information ( D ) 14 15 1. Submit one realworld application and the complete solution that each requires concepts in Calculus I in order to solve. 2. The realworld application and its solution must be original (your own work) and/or cited if other work was adapted. You may not simply copy or modify an existing application and/or solution. Note: Textbook word problems are NOT an acceptable applied project. 3. Use APA style to cite any references (websites, books, journals, etc...) that are used to inspire your problems (and/or solutions). 4. These realworld applications should be related to current events and should clearly demonstrate concepts in covered in our Calculus I. 5. Your grade for this project will be based on (a) originality, (b) accuracy of problem and its solution, (c) relevancy to current events, and (d) relevance and clear connection to concepts addressed in our Calculus I course. Below is an example without solution. Need Solution in project. Focus on Calculus in Civil Engineering Suppose you are a civil engineer. In 1919, the world's longest cantilever bridge (Pont de Qubec) was finally completed after several failed attempts. The center cantilever span of 1800 feet remains the longest cantilevered bridge span in the world. Suppose the Canadian National Railway wants you to describe the following key physical traits of the bridge (curvature, shear force, height, and slope of the bridge span). (a) If x represents the position along the cantilever bridge span (with x=0 being the left-hand endpoint of the span), match the calculus expression with the appropriate physical interpretation of the cantilever bridge span. I. w(x) II. w ' (x) B. The shear force of the bridge span (at III. w' '(x) C. The height of the bridge span (at IV. w' ' ' (x) D. The slope of the bridge span (at A. The curvature of the bridge span (at x ). x ). x ). x ). (b) Describe each of the physical interpretations in your own words and construct figures to illustrate your reasoning. \f\f