Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises: 30

Answer

$$\frac{\partial}{\partial \alpha}\int_\alpha^\beta\sqrt{t^3+1}dt=-\sqrt{\alpha^3+1};$$ $$\frac{\partial}{\partial \beta}\int_\alpha^\beta\sqrt{t^3+1}dt=\sqrt{\beta^3+1}.$$

Work Step by Step

We will use the fact that $$\frac{d}{dx}\int_a^xf(t)dt=f(x)$$ if $a$ is a constant. Regard $\beta$ as constant and differentiate with respect to $\alpha$: $$\frac{\partial}{\partial \alpha}\int_\alpha^\beta\sqrt{t^3+1}dt=\frac{\partial}{\partial \alpha}\left(-\int_\beta^\alpha\sqrt{t^3+1}dt\right)=-\frac{\partial}{\partial \alpha}\left(\int_\beta^\alpha\sqrt{t^3+1}dt\right)=-\sqrt{\alpha^3+1}.$$ Regard $\alpha$ as constant and differentiate with respect to $\beta$: $$\frac{\partial}{\partial \beta}\int_\alpha^\beta\sqrt{t^3+1}dt=\sqrt{\beta^3+1}.$$
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