Calculus I, Fall 2014

Midterm Exam 1

Oct 23, 2014, 10:10AM

1. (10 pts) Find lim_{y→+∞}y sin \frac{2}{\sqrt{y}}.

2. (12 pts) Give formal definition of lim_{x→0^{+}} \frac{1}{x} = ∞ verify it using the ε − δ argument.

3. (12 pts) Find dy/dx where y = x^{(x^{x})}, x > 0. Need not simplify your expression.

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

5. (12 pts)

Find the smallest n such that \frac{d^{n}}{dx^{n}} (x^{10} sin x)|_{x=0} is nonzero and find this value.

6. (16 pts) True or False? If true, prove it. If false, give a counter example. If |f(x) − (3x + 2)| ≤ |x|^{1.5} for all x ∈ R, then f is differentiable at x = 0.

7. (12 pts) Write down L(x, x_{0}), the linear approximation of f near x_{0}. Find an approximate value of sin(\frac{π}{3} −0.01) such that the error of the approximation is smaller than 5×10^{−5}. (Hint: choose x_{0} carefully)

8. (16 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.