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So I'm given:

$h(1) = -2$
h'(1) = 2
h''(1) = 3
$h(2) = 6$
h'(2) = 5
h''(2) = 13

The question is find \int_{1}^{2} h''(u) \text{d}u

So based on the Fundamental Theorem of Calculus (part 2) $\int_{a}^{b} f(x)\text{d}x = F(b) - F(a)$

I would figure that : h''(u) = h'(2) - h'(1) == 3

Is that the right way of looking at this?

Thanks for any insight.

  • 0
    Don't you mean $\int_{1}^{2} h''(u) \text{d}u = h'(u)|_1^2 = h'(2) - h'(1) = 3$?2011-07-02

2 Answers 2

1

Given that h''(u) is continuous, you have it right.

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$h(x) = a x^3 + b x^2 + c x +d $

$h^{\prime\prime}(x) = 6ax + 2b$ to find $a$ and $b$ solve the equations $h^{\prime\prime}(1) = 3$ and $h^{\prime\prime}(2) = 13$

now

$\int_1^2h^{\prime\prime}(x)dx = \left[3ax^2+2bx\right]_1^2 $

  • 0
    Where exactly did $h(x) = ax^3 + bx^2 + cx + d$ come from? Never mind, I see how you worked backwards. However, I think that's a little too involved for the question, as asked in the book. The reason I think that is because the question before it, which was similar, has an answer in the back which correlates to how I solved it above.2011-03-31