4. (6 points) Express the domain of the following function in interval notation:

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1 Eam 1-A L. Ballou Name Math 131 Calculus I September 1, 016 NO Calculator Allowed BOX YOUR ANSWER! Show all work for full credit! 1. (4 points) Write an equation of a line with y-intercept 4 and -intercept -1.. (6 points) Solve the equation for : (6 points) Solve the equation for : 1 4. Epress your answer in interval notation (6 points) Epress the domain of the following function in interval notation:.

2 5. (8 points) Sketch a eample of a function with the following conditions: f 1, 0,,, f 1 0, 6. (6 points) If 5, must f defined at 1? If it is, must 1 anything about f 1? Eplain. f 1 5? Can we conclude 7. (6 points) Show that the equation has at least one solution on the interval 4,4.

3 8. (8 points) For the function graphed below, find the following its or eplain why they do not eist: a. 0 b (16 points) Suppose that f and g are functions with f ( ) 5, g( ). a. Find c f g c c Show your work and state which it laws (see attached) were used. b. Find 3 c g Show your work and state which it laws (see attached) were used.

4 10. (6 points each) Find the following its, if they eist. Eplain. a. 710 b c d. f( ) if 3 3 i 3 f ( ) 0 i 3 1 i 3 b i (10 points) Let f ( ) 4 i 1 c 3 i 1 the definition of continuity in your answer., find c and b such that f is continuous at 1.Use

5 Limit Laws Suppose that k is a constant and the its f ( ) and g( ) c c eist. Then 1. Constant Multiple: kf ( ) k f ( ) c c. Sum/Difference Rule: 3. Product rule: f ( ) g( ) f ( ) g( ) c c c f ( ) g( ) f ( ) g( ) c c c f( ) f( ) c 4. Quotient rule: c g ( ) if g ( g ( ) 0 ) a c 5. Composition Rule: f g f g c c 6. Cancellation Theorem for Limits -- If g, if f is continuous at g c c eists and f is a function that is equal to g for all sufficiently close to c ecept possibly at c itself, then g c c 7. The Squeeze Theorem for Limits If l u for all sufficiently close to c, but not necessarily at c, and if l L u, then L c c 8. Limits Whose Denominators Approach Zero from the Right or the Left f a. If is of the form 1 c g 0, then f c g b. If c g is of the form 1 0, then c g c 9. Limits Whose Denominators Become Infinite Approach Zero f a. If is of the form 1 g, then f 0 g b. If sin 1 g 1 cos 0 is of the form 1, then and 1 1/ 0 f g 0 e Similar results hold for its at infinity and one-sided its

6 Eam 1-B L. Ballou Name Math 131 Calculus I September 1, 016 NO Calculator Allowed BOX YOUR ANSWER! Show all work for full credit! 1. (4 points) Write an equation of a line with slope 5 and y-intercept (6 points) Solve the equation for : (6 points) Solve the equation for : Epress your answer in interval notation (6 points) Epress the domain of the following function in interval notation:.

7 5. (8 points) Sketch a eample of a function with the following conditions: 1, 1, 1,, 1, and 0 f (6 points) If f 1 5, must anything about 1 1? Eplain. eist? If it does, them must 1 5? Can we conclude 7. (6 points) Show that the equation has at least one solution on the interval,.

8 8. (8 points) For the function graphed below, find the following its or eplain why they do not eist: a. 0 b (16 points) Suppose that f and g are functions with f ( ) 5, g( ). a. Find c f g c c Show your work and state which it laws (see attached) were used. b. Find 3 used. c g Show your work and state which it laws (see attached) were

9 10. (6 points each) Find the following its, if they eist. Eplain. 3 a. 1 b c d. 3 if 3 i 3 f ( ) 0 i 3 1 i 3 b i (10 points) Let f ( ) 4 i 1 c 3 i 1 the definition of continuity in your answer., find c and b such that f is continuous at 1.Use

10 Limit Laws Suppose that k is a constant and the its f ( ) and g( ) c c eist. Then 11. Constant Multiple: kf ( ) k f ( ) c c 1. Sum/Difference Rule: 13. Product rule: f ( ) g( ) f ( ) g( ) c c c f ( ) g( ) f ( ) g( ) c c c f( ) f( ) c 14. Quotient rule: c g ( ) if g ( g ( ) 0 ) a c 15. Composition Rule: f g f g c c 16. Cancellation Theorem for Limits -- If g, if f is continuous at g c c eists and f is a function that is equal to g for all sufficiently close to c ecept possibly at c itself, then g c c 17. The Squeeze Theorem for Limits If l u for all sufficiently close to c, but not necessarily at c, and if l L u, then L c c 18. Limits Whose Denominators Approach Zero from the Right or the Left f a. If is of the form 1 c g 0, then f c g b. If c g is of the form 1 0, then c g c 19. Limits Whose Denominators Become Infinite Approach Zero f a. If is of the form 1 g, then f 0 g 0. 0 b. If sin 1 g 1 cos 0 is of the form 1, then and 1 1/ 0 f g 0 e Similar results hold for its at infinity and one-sided its

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