Calculus question: We are given that f(3) = 4, f′(3) = 8, f′′(3) = 11. Use this information to approximate f(3.15).

Question

joannablackwelder · Answer

Is there a better method of doing this than just using Newton's method and ignoring the second derivative info?

joannablackwelder · Answer

@IrishBoy123

irishboy123 · Answer

Joanna, i would go with the linear approximations you suggest.

joannablackwelder · Answer

Thanks! @UnkleRhaukus Any thoughts?

inkyvoyd · Answer

do a quadratic approximation

inkyvoyd · Answer

f''(x) will be constant near f(x)

joannablackwelder · Answer

Oh, I see, I found a formula for that. Thanks so much!

inkyvoyd · Answer

... and I deleted it because it's not right. Sigh, been so long since I've used calc 2. Anyways, fun fact, the taylor series is basically a continuation of this... https://en.wikipedia.org/wiki/Taylor_series#Calculation_of_Taylor_series

unklerhaukus · Answer

Assuming $$f(x) = ax^2+bx+c$$$$f'(x) = 2ax+b$$$$f''(x) = 2a$$

joannablackwelder · Answer

unklerhaukus · Answer

$$f''(3) = 2a =11\implies a=11/2$$

joannablackwelder · Answer

Using the quadratic approximation that I linked to before, I get about 5.32

unklerhaukus · Answer

$$f'(3)=8=11(3)+b\implies b=-25$$
$$f(3)=4=\tfrac{11}2(3)^2-25(3)+c\implies c=29.5$$

unklerhaukus · Answer

$$f(x) = \tfrac{11}2x^2-25x+29.5$$

$$f(3.15) = \tfrac{11}2(3.15)^2-25(3.15)+29.5 = 5.32375$$

freckles · Answer

I wonder if taylor series is what they want
$$f(x) \approx f(a)+f'(a)(x-a)+\frac{f'(a)}{2}(x-a)^2 \text{ where } a \text{ here equals } 3$$

freckles · Answer

oh you mentioned that formula in that link @joannablackwelder nevermind

joannablackwelder · Answer

Thanks so much everybody!