Chapter 4: Partial Differentiation

Q21MP

Page 239

Find the hottest and the coldest point on the bar graph of 5. If $\mathit{T}\mathbf{=}\mathbf{4}\mathit{x}\mathbf{-}{\mathit{x}}^{\mathbf{2}}$, where x is distance measured from the left end.

Q22P

Page 211

If$\mathit{w}\mathbf{=}\mathit{f}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathit{y}\mathbf{)}$, show that$\mathit{b}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{x}}\mathbf{-}\mathit{a}\frac{\mathbf{\partial }\mathbf{w}}{\mathbf{\partial }\mathbf{y}}\mathbf{=}\mathbf{0}$

Q22P

Page 191

If,$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathbf{}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{,}\mathbf{}\mathit{y}\mathbf{=}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }$, find the following partial derivatives.

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }\mathbf{r}\mathbf{\partial }\mathbf{x}}$.

Q23MP

Page 239

Find $\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{t}}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{-}\mathbf{1}}\mathbf{x}}{\mathbf{x}}\mathit{d}\mathit{x}$.

Q23P

Page 211

If $\mathit{u}\mathbf{=}\mathit{f}\mathbf{(}\mathit{x}\mathbf{-}\mathit{c}\mathit{t}\mathbf{)}\mathbf{+}\mathit{g}\mathbf{(}\mathit{x}\mathbf{+}\mathit{c}\mathit{t}\mathbf{)}$, show that $\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{u}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{c}}^{\mathbf{2}}}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{u}}{\mathbf{\partial }{\mathbf{t}}^{\mathbf{2}}}\mathbf{.}$

Q24MP

Page 239

Find $\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathbf{\int }}_{\mathbf{t}\mathbf{=}\frac{\mathbf{1}}{\mathbf{x}}}^{\mathbf{t}\mathbf{=}\frac{\mathbf{2}}{\mathbf{x}}}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{x}\mathbf{t}}{\mathbf{t}}\mathit{d}\mathit{t}$.

Q25MP

Page 239

Find $\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathbf{\int }}_{\mathbf{1}}^{\frac{\mathbf{1}}{\mathbf{x}}}\frac{{\mathbf{e}}^{\mathbf{x}\mathbf{t}}}{\mathbf{t}}\mathit{d}\mathit{t}$.

Q26MP

Page 239

Find $\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathbf{\int }}_{\mathbf{0}}^{{\mathbf{x}}^{\mathbf{2}}}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}\mathbf{t}}{\mathbf{t}}\mathit{d}\mathit{t}$.

Q26P

Page 211

Given $\mathit{f}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\mathbf{0}$and $\mathit{g}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,}$find a formula for $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$.

Q27MP

Page 239

Show that $\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathbf{\int }}_{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{x}}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\mathit{d}\mathit{t}\mathbf{=}\mathbf{1}$.