Calculus I, Fall 2014

Brief solutions to Midterm Exam 1

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

Ans:

= lim_{y→+∞} \sqrt{y} 2 \frac{\sqrt{y}}{2} sin \frac{2}{\sqrt{y}} = ∞ · 2 · 1 = ∞.

2. (12 pts) Give formal definition of lim_{x→0^{+}} \frac{1}{x} = ∞ and verify it using the ε − δ argument. Ans: For any M ∈ R, there exist δ > 0 such that 0 < x < δ implies f(x) > M. Proof: If M > 0, take δ = 1/M. If M ≤ 0, take any δ > 0 (for example δ = 1) will do. Then verify (details skipped).

3. (12 pts) Find dy/dx where y = x^{(x^{x})}, x > 0. Need not simplify your expression. Ans: = x^{(x^{x})}x^{x}(ln x(1 + ln x) + \frac{1}{x}).

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. Ans: \frac{d}{dx} once:

\frac{1 + y′}{1 + (x + y)^{2}} + \frac{2x + y′}{\sqrt{1 − (x^{2} + y)^{2}}} = 0,

evaluate at x = 1, y = −1, one gets y′(1, −1) = −\frac{3}{2}. \frac{d}{dx} twice:

\frac{y′′(1 + (x + y)^{2}) − 2(x + y)(1 + y′)^{2}}{(1 + (x + y)^{2})^{2}} + \frac{(2 + y′′)(1 − (x^{2} + y)^{2}) + (x^{2} + y)(2x + y′)^{2}}{(1 − (x^{2} + y)^{2})^{\frac{3}{2}}} = 0

evaluate at x = 1, y = −1, one gets y′′(1, −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.

Ans: n = 11. \frac{d^{11}}{dx^{11}} (x^{10} sin x)|_{x=0} = 11!.

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. Ans: True. Step 1: let x = 0, we have f(0) = 2.

Step 2:

lim_{x→0} \frac{f(x) − f(0)}{x − 0} = lim_{x→0} \frac{f(x) − (3x + 2)}{x − 0} + 3.

Since |\frac{f(x)−(3x+2)}{x−0}| ≤ |x|^{0.5}, it follows from Sandwich Theorem that

lim_{x→0} \frac{f(x) − (3x + 2)}{x − 0} = 0.

Therefore

lim_{x→0} \frac{f(x) − f(0)}{x − 0} = 3, differentiable.

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) Ans: L(x, x_{0}) = f(x_{0}) + f′(x_{0})(x − x_{0}). Choose f(x) = sin x, x_{0} = \frac{π}{3}. sin(\frac{π}{3} − 0.01) ≈ sin(\frac{π}{3}) + cos(\frac{π}{3})(−0.01) = \frac{\sqrt{3}}{2} − 0.005.

|Error| ≤ \frac{1}{2}| sin c| 0.01^{2} ≤ 5 × 10^{−5}

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. Ans: Let y = f(x), then x = f^{−1}(y) and

f^{−1}(y) = f^{−1}(f(x)) = x, or f(x) = f(f^{−1}(y)) = y

take \frac{d}{dx} twice on the first equation or \frac{d}{dy} twice on the second equation both lead to

\frac{d^{2}}{dy^{2}} f^{−1}(y)(f′(x))^{2} + \frac{d}{dy} f^{−1}(y)f′′(x) = 0

which gives

\frac{d^{2}}{dy^{2}} f^{−1}(y)|_{y=f(x)} = −\frac{f′′(x)}{(f′(x))^{3}} ,

or

\frac{d^{2}}{dy^{2}} f^{−1}(y) = −\frac{f′′ (f^{−1}(y))}{(f′(f^{−1}(y)))^{3}} ,