Chapter 5: Applications of Derivatives

Question 1

Linear Approximations

Answer 1

Suppose y=f(x) is differentiable
f(x)-f(x0)=f’(x0)(x-x0)+o(x-x0) where a(x) is an infinitesimal as x approaches x0.
f(x)-f(x0)=f’(x0)(x-x0) where x is near x0

Question 2

Differential

Answer 2

Let y=f(x) be a differentiable function. The differential of the function is defined by
dy=df=f’(x)deltax

Question 3

Higher Differentials

Answer 3

Let y=f(x) have derivatives until the nth-degree then we have the definition of higher differentials:
d^2y=d(dy)=d^2f=d(df)=f’(x)dx=f’‘(x)d^2x.
Taylor Polynomials(Extended)
ck=(f(x0)(k)/k!).(x-x0)^n

Question 4

The Mean Value Theorems

Answer 4

Ferma’s Theorem:
- If f(x) has a local minimum or maximum at C,if f’(C) exists then f’(C)=0 (C is considered to be the critical point)
Rolle’s Theorem:
- As long as f is differentiable on [a,b] and continuous on (a,b) then f(a)=f(b)
Lagrange’s Theorem:
- With the same condition as Rolle’s Theorem Subsequently, we have a number of c belongs to (a,b) such that:
f’(c)=(f(b)-f(a))/(b-a)
Corollary’s Theorem:
f(x)=C for all x belongs to (a,b) as long as f’(x)=0.
L’Hospital’s Rule:
Suppose that f and g are differentiable and g’(x) is different than 0 then we have:
lim (f(x)/g(x))=lim (f’(x)/g’(x)) as x tends to a.
REMEMBER: The above equation is possible only if limf(x)=0 and lim g(x)=0 or lim f(x)= infinity and lim g(x)= infinity as x tends to a.