Recall that linear approximation to the function f(x) at a is given by L(x)= f(a)+f'(a)(x-a)
1. Setting f(x)=x^4 and a=2 give an estimate of (1.999)4.
2. A quadratic approximation is given by Q(x)=f(a)+f'(a)(x-a)+1/2 f''(a)(x-a)^2. Use this formula to approximate (1.999)^4.
3. Use a calculator to determine the true value of (1.999)^4. Which approximation was better, the linear or quadratic?
4. Guess a formula for a cubic approximation.

Question

Recall that linear approximation to the function f(x) at a is given by L(x)= f(a)+f'(a)(x-a)
1. Setting f(x)=x^4 and a=2 give an estimate of (1.999)4.
2. A quadratic approximation is given by Q(x)=f(a)+f'(a)(x-a)+1/2 f''(a)(x-a)^2. Use this formula to approximate (1.999)^4.
3. Use a calculator to determine the true value of (1.999)^4. Which approximation was better, the linear or quadratic?
4. Guess a formula for a cubic approximation.

anonymous · Answer

$f'(x)=4x^3$ so $f'(2)=32$

anonymous · Answer

if $a=2$ then $f(a)=2^4=16$ and so $$L(x)=16+32(x-2)$$ is the linear approximation

anonymous · Answer

if you put $L(1.999)$ you get 
$$16+32\times (-.001)$$