Section 1
Functions of several variables
If$$f(x, y)=5 x+x y^{2}-10$$and$$g\left(x_{1}, x_{2}, x_{3}\right)=x_{1}+x_{2}+x_{3}$$evaluate(a) $f(0,0)$(b) $f(1,2)$(c) $f(2,1)$(d) $g(5,6,10)$(e) $g(0,0,0)$(f) $g(10,5,6)$
Find expressions for the first-order partial derivatives for the functions(a) $f(x, y)=5 x^{4}-y^{2}$(b) $f(x, y)=x^{2} y^{3}-10 x$
Find expressions for the second-order partial derivatives of the functions(a) $f(x, y)=5 x^{4}-y^{2}$(b) $f(x, y)=x^{2} y^{3}-10 x$[Hint: you might find your answer to Practice Problem 2 useful.]
Find expressions for the partial derivatives $f_{1}, f_{11}$ and $f_{21}$ in the case when$$f\left(x_{1}, x_{2}, x_{3}\right)=x_{1} x_{2}+x_{1}^{5}-x_{2}^{2} x_{3}$$
If$$z=x y-5 x+2 y$$evaluate$\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$at the point $(2,6)$.(a) Use the small increments formula to estimate the change in $z$ as $x$ decreases from 2 to $1.9$ and $y$ increases from 6 to $6.1$.(b) Confirm your estimate of part (a) by evaluating $z$ at $(2,6)$ and (1.9, 6.1).
Use implicit differentiation to find expressions for $\mathrm{d} y / \mathrm{d} x$ given that(a) $x y-y^{3}+y=0$(b) $y^{5}-x y^{2}=10$
If$$f(x, y)=3 x^{2} y^{3}$$evaluate $f(2,3), f(5,1)$ and $f(0,7)$.
If$$f(x, y)=2 x y+3 x$$verify that $f(5,7) \neq f(7,5)$. Are there any pairs of numbers, $(x, y)$ for which $f(x, y)=f(y, x) ?$
Write down expressions for the first-order partial derivatives, $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for(a) $z=x^{2}+4 y^{5}$(b) $z=3 x^{3}-2 \mathrm{e}^{y}$(c) $z=x y+6 y$(d) $z=x^{6} y^{2}+5 y^{3}$
If$$f(x, y)=x^{4} y^{5}-x^{2}+y^{2}$$write down expressions for the first-order partial derivatives, $f_{x}$ and $f_{y}$. Hence evaluate $f_{x}(1,0)$ and $f_{y}(1,1)$
Find expressions for all first- and second-order partial derivatives of the following functions. In each case verify that$$\frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial^{2} z}{\partial x \partial y}$$(a) $z=x y$(b) $z=\mathrm{e}^{x} y$(c) $z=x^{2}+2 x+y$(d) $z=16 x^{1 / 4} y^{3 / 4}$(e) $z=\frac{y}{x^{2}}+\frac{x}{y}$
Use the small increments formula to estimate the change in$$z=x^{2} y^{4}-x^{6}+4 y$$when(a) $x$ increases from 1 to $1.1$ and $y$ remains fixed at 0(b) $x$ remains fixed at 1 and $y$ decreases from 0 to $-0.5$(c) $x$ increases from 1 to $1.1$ and $y$ decreases from 0 to $-0.5$
If$$z=x^{2} y^{3}-10 x y+y^{2}$$evaluate $z_{x}$ and $z_{y}$ at the point $(2,3)$. Hence estimate the change in $z$ as $x$ increases by $0.2$ and $y$, decreases by $0.1$.
(a) If$$f(x, y)=y-x^{3}+2 x$$write down expressions for $f_{x}$ and $f_{y}$. Hence use implicit differentiation to find dy/dx given that$$y-x^{3}+2 x=1$$(b) Confirm your answer to part (a) by rearranging the equation$$y-x^{3}+2 x=1$$to give y explicitly in terms of $x$ and using ordinary differentiation.
Verify that $x=1, y=-1$ satisfy the equation $x^{2}-2 y^{3}=3$. Use implicit differentiation to find the value of $\mathrm{d} y / \mathrm{d} x$ at this point.
A function of three variables is given by$$f\left(x_{1}, x_{2}, x_{3}\right)=\frac{x_{1} x_{3}^{2}}{x_{2}}+\ln \left(x_{2} x_{3}\right)$$Find all of the first- and second-order derivatives of this function and verify that $f_{12}=f_{21}, \quad f_{13}=f_{31} \quad$ and $f_{23}=f_{32}$