11.2.1: Partial Derivatives (Exercises)
- Page ID
- 104879
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the following exercise, calculate the partial derivative using the limit definitions only.
1) \(\dfrac{∂z}{∂y}\) for \( z=x^2−3xy+y^2\)
- Answer
- \(\dfrac{∂z}{∂y}=−3x+2y\)
In exercises 2 - 13, calculate the requested partial derivatives.
2) \( \dfrac{∂z}{∂x}\) for \( z=\sin(3x)\cos(3y)\)
3) \( \dfrac{∂z}{∂y}\) for \( z=\sin(3x)\cos(3y)\)
- Answer
- \( \dfrac{∂z}{∂y}=−3\sin(3x)\sin(3y)\)
4) \( \dfrac{∂z}{∂x}\) and \( \dfrac{∂z}{∂y}\) for \( z=x^8e^3y\)
5) \( \dfrac{∂z}{∂x}\) and \( \dfrac{∂z}{∂y}\) for \( z=\ln(x^6+y^4)\)
- Answer
- \( \dfrac{∂z}{∂x}=\frac{6x^5}{x^6+y^4};\quad \dfrac{∂z}{∂y}=\frac{4y^3}{x^6+y^4}\)
6) Find \( f_y(x,y)\) for \( f(x,y)=e^{xy}\cos(x)\sin(y).\)
7) Let \( z=e^{xy}.\) Find \( \dfrac{∂z}{∂x}\) and \( \dfrac{∂z}{∂y}\).
- Answer
- \( \dfrac{∂z}{∂x}=ye^{xy};\quad \dfrac{∂z}{∂y}=xe^{xy}\)
8) Let \( z=\ln(\frac{x}{y})\). Find \( \dfrac{∂z}{∂x}\) and \( \dfrac{∂z}{∂y}\).
9) Let \( z=\tan(2x−y).\) Find \( \dfrac{∂z}{∂x}\) and \( \dfrac{∂z}{∂y}\).
- Answer
- \( \dfrac{∂z}{∂x}=2\sec^2(2x−y),\quad \dfrac{∂z}{∂y}=−\sec^2(2x−y)\)
10) Let \( z=\sinh(2x+3y).\) Find \( \dfrac{∂z}{∂x}\) and \( \dfrac{∂z}{∂y}\).
11) Let \( f(x,y)=\arctan(\frac{y}{x}).\) Evaluate \( f_x(2,−2)\) and \( f_y(2,−2)\).
- Answer
- \( f_x(2,−2)=\frac{1}{4}=f_y(2,−2)\)
12) Let \( f(x,y)=\dfrac{xy}{x−y}.\) Find \( f_x(2,−2)\) and \( f_y(2,−2).\)
13) Find \( \dfrac{∂z}{∂x}\) at \( (0,1)\) for \( z=e^{−x}cos(y)\).
14) Express the volume of a right circular cylinder as a function of two variables:
a. its radius \( r\) and its height \( h\).
b. Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height.
c. Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.
- Answer
- \( a. V(r,h)=πr^2h\)
\( b. \dfrac{∂V}{∂r}=2πrh\)
\( c. \dfrac{∂V}{∂h}=πr^2\)
15) The function \( P(T,V)=\dfrac{nRT}{V}\) gives the pressure at a point in a gas as a function of temperature \( T\) and volume \( V\). The letters \( n\) and \( R\) are constants. Find \( \dfrac{∂P}{∂V}\) and \( \dfrac{∂P}{∂T}\), and explain what these quantities represent.