Calculus I, Fall 2015

Midterm Exam 1

Oct 27, 2015, 10:10AM

1. (10 pts) Evaluate lim_{θ→ \frac{π}{3}} \frac{cos θ − \frac{1}{2}}{θ − \frac{π}{3}} .

2. (10 pts) Let f_{i}(x) = a_{i}x^{2} + b_{i}x + c_{i}, i = 1, 2, 3. Evaluate

\frac{d}{dx} | \begin{matrix}f_{1}(x) & f′_{1}(x) & f′′_{1}(x) \\ f_{2}(x) & f′_{2}(x) & f′′_{2}(x) \\ f_{3}(x) & f′_{3}(x) & f′′_{3}(x) \end{matrix} |

3. (10 pts) Find \frac{dy}{dx} where y = x^{(x^{x})}, x > 0.

4. (10 pts) Find y′(1) and y′′(1) where y(x) is implicitly given by tan(x+y)+sin^{−1}(x^{2}+y) = 0 near (x, y) = (1, −1).

5. (10 pts) Let f^{−1} be the inverse function of f. Evaluate \frac{d^{2}}{dy^{2}} f^{−1}(y) in terms of f′ and f′′. Show all details.

6. (10 pts) Let f(x) : [0, 1] ↦ [0, 1] be a continuous function. Prove that f(x) = x has at least one solution.

7. (10 pts) Start with domain and range for csc and csc^{−1}, derive the formula for the derivative of csc^{−1}.

8. (10 pts) True or False? If true, prove it. If false, give a counter example. (a) If y = f(x) is differentiable at x = c then it is continuous at x = c.

(b) If y = f(x) is continuous at x = c then it is differentiable at x = c.

9. (10 pts) Give formal definition of lim_{x→∞}f(x) = L. Then verify that lim_{x→∞} \frac{1}{x} = 0 using the ε − δ argument.

10. (16 pts) Give formal definition of lim_{x→c} f(x) ≠ L. Then verify that lim_{x→0} sin x ≠ 1 using ε − δ argument.