Exam 2 Practice

Part I

(1) Calculate the derivatives of the following:

(a) $f(x) = {(x-\frac{1}{x^2})}^2 \sin x$

(b) $g(x) = \sec(x) \tan (x)$

(c) $h(x) = \frac{x^7}{{(x^2-4)}^2}$

(d) $f(x) = \sqrt{x^3 - \sin(x^2)}$

(e) $g(x) = \sin{(x^2+\frac{1}{x})}$

(f) $y = \frac{1}{\cot x}$

Part 2

(2) (a) Find $\frac{d^2y}{dx^2}$ for x²+y² = 16 in terms of x and y.

(b) What is $\frac{d^2y}{dx^2}$ when x= 2 and y > 0?

(3) Find the rate of change of the distance between the origin and a particle on the graph of y = x² + x if $\frac{dx}{dt} = 3$ and x = 2.

(4) Find the maximum and minimum values of the function f(x) = 5x⁴ - 10x³ - 15x² + 12 on [-1, 2].

(5) (a) Which of the following functions satisfy the hypotheses of the mean value theorem:

(i) f(x) = x² - 4x on [-1,3]

(ii) $g(x) = \csc x$ on [0, 2$\pi$]

(b) For those that fail the hypotheses state why.

(c) Give the value of c guaranteed to exist for the functions that do satisfy the mean value theorem.

(6) Let f(x) = 7x⁴ -14x³ -42x² +6x +9.

(a) Find the critical values of f(x).

(b) Where is f(x) increasing and where is it decreasing?

(c) State the relative maxima and minima of f(x).

(7) ?????

(8) ????