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Thread: higher-order derivatives

  1. #1
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    higher-order derivatives

    find a quadratic function f such that f(3)=33, f'(3)=22, f''(3)=8
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    find a quadratic function f such that f(3)=33, f'(3)=22, f''(3)=8
    start with $\displaystyle f(x) = ax^2 + bx + c$

    since $\displaystyle f(3) = 33$ ... $\displaystyle 33 = a(9) + b(3) + c$


    $\displaystyle f'(x) = 2ax + b$

    since $\displaystyle f'(3) = 22$ ... $\displaystyle 22 = 2a(3) + b$


    $\displaystyle f''(x) = 2a$

    since $\displaystyle f''(3) = 8$ ... $\displaystyle 8 = 2a$


    so ... find $\displaystyle a$, $\displaystyle b$, and $\displaystyle c$.
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  3. #3
    Super Member malaygoel's Avatar
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    Quote Originally Posted by skeske1234 View Post
    find a quadratic function f such that f(3)=33, f'(3)=22, f''(3)=8
    Assume $\displaystyle f(x)=ax^2+bx+c$

    Now, apply the given conditions and determine a,b,c.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Or you can use the Taylor series of the polynomial about $\displaystyle x=3$, and not have to solve for any unknowns:

    $\displaystyle f(x)=f(3)+\frac{f'(3)}{1!}(x-3)+\frac{f''(3)}{2!}(x-3)^2
    =33+22(x-3)+4(x-3)^2 = 4x^2-2x+3$
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