DIFFERENTIATION

Problems:

1. Show that if f is differentiable at a then f is continuous at a.

2. Give an example of a function f which is continuous at a point a but not differentiable at a.

3. f(x,y) = \{ \begin{matrix}ax^{2} + bx + 7 & if x < 1 \\ 2bx + 2a & if x ≥ 1 \end{matrix}

Suppose that f is differentiable at 1 Find a, b.

4. Find \frac{dy}{dx} for the following:

(1) y = sin^{3} (tan x)

(2) y = sec(cos^{\frac{1}{2}} (x^{2} + 5 − \frac{1}{x} ))

(3) y = x^{2} cot(csc x) + x sin x cos x

(4) y = sin(x^{2} + 1 ) + \frac{x + 2}{x^{3} + 1}

5. y = a_{n}x^{n} + a_{n−1} x^{n−1} + ⋯ + a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0} Find y′, y′′, y^{(n)} and y^{(n+1 )}.

6. Find \frac{dy}{dx} if y^{3} = x^{2} + cos \frac{x}{y^{2}} .

7. Find \frac{d^{2}y}{dx^{2}} if xy = sin x + cos y.

8. Let C denote the curve y^{4} − 4y^{2} = x^{4} − 9x^{2} .

(1) Show that (3,2) and (3,-2) are points on the curve C.

(2) Find the equation for the tangent to the curve C at (3,2).

(3) Find the equation for the normal to the curve C at (3,-2).

9. Let C be the curve described by

x = t − sin t, y = 1 − cos t

(1) Find \frac{dy}{dx} , \frac{d^{2}y}{dx^{2}} for t=\frac{π}{3} .

(2) Find the equation for the line tangent to the curve C at the point defined by t=\frac{π}{3} .

10. A, B are walking on the streets that meet at right angles. A approaches the intersection at 3 m/sec; B moves away from the intersection at 2 m/sec. At what rate is the angle θ changing and at what rate is the distance between A and B changing when A is 10 m from the intersection and B is 25 m from the intersection?

11. (1) f (x) = \sqrt[3]{x}

Find the differential at f .

(2) Use the differential to estimate \sqrt[3]{1000.2} .