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SECTION: (circle one) 11:00-11:55 12:05-1:
Math 105 - Exam 1 - October 7, 2005 Instructions: Show all of your work and circle your final answers. Calculators are allowed, but notes and books are not.
- (24 pts. - 8 pts. each) (a) Consider the differential equation dydt = y t+1−^2. Is y = 2t + 4 a solution? Justify your answer.
(b) Give all possible solutions of F to the differential equation F ′^ = 4x^2 + 3e^2 + 2x.
(c) If f (x) = (sin x)√x + (^) lnx x , find f ′(x).
- (10 pts.) State the formal limit definition of f ′(a).
- (24 pts. - 8 pts. each) Consider the function f (x) = x^2 − 4 x. (a) Calculate the average rate of change of f (x) on the interval [3, 4].
(b) Using your definition in Problem 2, calculate f ′(3).
(c) Find the equation of the tangent line on the graph of y = f (x) at the point where x = 3.
- (10 pts. - 5 pts. each) (a) Find the domain of the function g(x) = √ 4 − x^2.
(b) Suppose the range of the function f (x) is [− 1 , 1). Calculate the range of the function f (x + 3) + 4.
- BONUS (5 pts): Calculate limx→ 0 cos^ xx −^1 (and justify your answer).
Grading : do not write in this area
1 2 3 4 5 B TOTAL
