Calculus I, Fall 2013

Midterm Exam 1

Oct 29, 2012, 10:10AM

1. (8 pts) Find lim_{y→+∞}y sin \frac{2}{\sqrt{y}}.

2. (12 pts) Give precise definition of lim_{x→0^{+}} \frac{1}{x} = ∞ and show that it is true (using the ε − δ argument).

3. (8 pts) Find dy/dx where y = x^{x}, x > 0. Need not simplify your expression.

4. (12 pts) Find y′ and y′′ at (1, −1) where y(x) is implicitly given by tan(x + y) + sin(x^{2} + y) = 0.

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. (12 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. (12 pts) True or False? (prove it if true, correct it if false). Since x ↦ ln x and x ↦ e^{x} are inverse function to each other and \frac{d}{dx} ln x = \frac{1}{x}. Therefore \frac{d}{dx}e^{x} = \frac{1}{\frac{d}{dx} ln x} = \frac{1}{\frac{1}{x}} = x.

9. (12 pts) Find absolute maximum and absolute mininum of f(x) = x^{1/3}(x − 1/2) on [−1, 1].