We are given that f(3) = 4, f′(3) = 8, f′′(3) = 11. Use this information to approximate f(3.15).
Is f increasing at x = 3?

Question

We are given that f(3) = 4, f′(3) = 8, f′′(3) = 11. Use this information to approximate f(3.15).
Is f increasing at x = 3?

calculusfunctions · Answer

Sorry, had to leave for a second. Are you ready?

calculusfunctions · Answer

Do you know the quadratic approximation to f(x)?

anonymous · Answer

I can calculate y= f'(3) (x-3) + f(3), so if f'(3) = 8 and f(3) = 4, we would write as 8(x-3)+4... but I can't figoure out how to account for the second derivative

calculusfunctions · Answer

$$f(x)\approx f(a)+f \prime(a)(x -a)+\frac{ f \prime \prime(a) }{ 2 }(x -a)^{2}$$Do you know this formula?

anonymous · Answer

hrm... no, I am not familiar with that formula.

calculusfunctions · Answer

That formula is the quadratic approximation to f(x) near x = a.  What you did @husbishag is use the linear approximation to f(x) near x = a. That won't work because we have f ''(3). Understand?

anonymous · Answer

I believe I understand, so would we write this out as 4+8(x-3) + 11/2(x-3)^2, and if we approximate f at3.15, than 4+8(3.15-3)+11/2(3.15-3)^2 = 5.32???

calculusfunctions · Answer

Exactly! You got it dude! Great Job!!! I haven't checked with a calculator if it yields approx. 5.32 but the left side is correct!! @husbishag do you understand now?

anonymous · Answer

YES!!! I get it! Thank you so much for your help! I appreciate you walking me through the answer and not just providing it! THANK YOU!!

calculusfunctions · Answer

Well I'm a teacher so therefore never count on me to just give you the answer, because teach, is what I do!!!!!! No need for thanks, as long as you've learned something!

calculusfunctions · Answer

I just calculated it and yes it is approx. 5.32.

calculusfunctions · Answer

If that's all for now, I have to go.

anonymous · Answer

@calculusfunctions, while I have your attention, can I ask you a question? I'm very new to calculus and my understanding is on the fringe at best. I have a question that asks compute the derivative of f(x) = 5 + 3x + 6x^2 = 7x^3, would I record this as f'(x) = 3+12x+21x^2? Is 5 considered a constant, and therefore zero?

calculusfunctions · Answer

Yes that would be the correct derivative of that function! @husbishag have more faith in yourself, you're doing great!!!!!

anonymous · Answer

Alright... thank you again. Maybe I am 'getting it' more than I'm giving myself credit. thanks.

calculusfunctions · Answer

Definitely take care! I have to go now but if you need anything else from me, let me know. Take care, Bye!

anonymous · Answer

Thanks! Have a good day!

calculusfunctions · Answer

Thanks you too!