Derivatives 2

Question 1

What is the chain rule

Answer 1

Chain rule is:
If there are continuous functions g,f (A,B) to R then derivative of fog is f’(g(c)) * g’(c) provided g is differentiable at c and f is differentiable at g(c)

Question 2

If f is bijection and differentiable at c, when is f^-1 differentiable

Answer 2

If f is a bijection and differentiable at c, f^-1 is continuous at f(c) and:
F^-1(y0) = 1/f’(f^-1(y0)) = 1/f’(c)

Question 3

If f is increasing or decreasing in interval, what is f’(x)

Answer 3

If f is increasing on interval, f’(x) >= 0 (reciprocal false in general, x^3 in [1,-1]
If decreasing f’(x) <= 0

Question 4

If f’(x) > 0 or < 0, what does this imply

Answer 4

If f’(x) > 0, this implies f is strictly increasing

If < 0 f is strictly decreasing

Question 5

When is function f n-times differentiable at c and examples

Answer 5

Function f is differentiable at c is:
F is (n-1)-times differentiable at all points in (a,b)
F^n-1 (n-1th derivative) is differentiable at c
Examples are sim, cos, polynomials

Question 6

What is the formula for the Taylor series

Answer 6

Formula for the Taylor series is:

F(x) = sum j=0 to n f^j(X0)/j! *(x-X0)^j + Rn(x)

Question 7

What is Rj(x) in Taylor series

Answer 7

Rn(x) in Taylor series is:

Rj(x) = f^(j+1) (c)/(j+1)! * (x - X0)^j+1

Question 8

What is the Taylor polynomial of f at X0 of order n

Answer 8

Taylor polynomial of f at X0 of order n is:
F(X0) + f’(X0)*(x - X0) as this is a polynomial of degree at most n
This comes from sum j=0 to n f^j(X0)/j! *(x-X0)