Calculus
1 Hour Exam 1 (continued) - Math 221
Fall, 2011
5. |
a. State the Intermediate Value Theorem. Be sure to include hypotheses and conclusion. b. Let f(x) = x3 – 3x + 1. Use this theorem to show that there is a number c between 1 and 2 such that f(c) = 0. You should write sentences (in addition to mathematical expressions) and be sure to verify, in writing, the hypotheses of this theorem. Answers: If f is continuous on [a,b] and N is between f(a) and f(b), then there Is a c ε [a,b] with f(c) = N. f is continuous since it is a polynomial; f(1) = -1 and f(2) = 3; 0 is between f(1) and f(2). So by this theorem there is a c ε (1,2) with f(c) = 0. |
6. |
Find dy/dx for y = 2 cos x. Then find the equation of the tangent line to the graph of y = 2 cos x at the point (0,2). Answers: dy/dx = - 2 sin x and y – 2 = 0 |
7. |
For this problem answer true or false for each part. You do not need to show work or give any reason. There is no partial credit on this problem. a. If lim [f(x) – f(1)] / (x-1) = 3, then lim f(x) = f(1). x→1 x→1 b. If f(x) ≤ g(x) ≤ h(x) for all x and if lim f(x) = 3 and if lim h(x) = 4, then x→2 x→2 lim g(x) exists and 3 ≤ g(x) ≤ 4. x→2 c. Polynomials are both continuous and differentiable for all x. d. If f(x) and g(x) are differentiable, then d/dx [f(x) g(x)] = f ‘(x) g ‘(x). e. If f(x) and g(x) are differentiable and if c is a constant, then d/dx [c f(x) – g(x)] = c f ‘(x) – g ‘(x).
Answers: a. T b. F c. T d. F e. T |