Chap 24 Review

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Exercise 1 What is $\partial_x x$?

Exercise 2 What is $\partial_x y$?

Exercise 3 What is $\partial_x a\, x$?

Exercise 4 What is $\partial_x x\, y$?

Exercise 5 What is $\partial_y x\, y$?

Exercise 6 What is $\partial_x A e^{kt}$?

$0$ $A k e^{kx}$ $t$

question id: drill-partialderivatives-6

Exercise 7 What is $\partial_t A e^{kt}$?

Exercise 8 What is $\partial_x A x e^{kt}$?

Exercise 9 What is $\partial_t A x e^{kt}$?

Exercise 10 What is $\partial_x \left[\strut a_0 + a_1 x + a_2 x^2 \right]$?

Exercise 11 What is $\partial_y \left[\strut a_0 + a_1 x + a_2 x^2 \right]$?

Exercise 12 What is $\partial_x \left[\strut a_0 + a_1 y + a_2 y^2 \right]$?

Exercise 13 What is $\partial_x \left[\strut a_0 + a_1 x + b_1 y + c x y \right]$?

Exercise 14 What is $\partial_y \left[\strut a_0 + a_1 x + b_1 y + c x y \right]$?

Exercise 15 What is $\partial_x \partial_y \left[\strut a_0 + a_1 x + b_1 y + c x y \right]$? (Usually we would write $\partial_{xy}$ instead of $\partial_x \partial_y$, but they amount to the same thing.)

Exercise 16 What is $\partial_x \partial_x \left[\strut a_0 + a_1 x + b_1 y + c x y \right]$? (Usually we would write $\partial_{xx}$ instead of $\partial_x \partial_x$, but they amount to the same thing.)

Exercise 17 What is $\partial_x \partial_x \left[\strut a_0 + a_1 x + b_1 y + c x y + a_2 x^2 + b_2 y^2 \right]$? (Usually we would write $\partial_{xx}$ instead of $\partial_x \partial_x$, but they amount to the same thing.)

Exercise 18 What is $\partial_y \partial_x \left[\strut a_0 + a_1 x + b_1 y + c x y + a_2 x^2 + b_2 y^2 \right]$? (Usually we would write $\partial_{yx}$ instead of $\partial_y \partial_x$, but they amount to the same thing.)

Exercise 19 What is $\partial_x \left[\strut A x^n y^m \right]$?

$A n x^{n-1} y^m$ $A n m x^{n-1} y^{m-1}$ $A m x^{n} y^{m-1}$ $A y^m$

question id: drill-partialderivatives-19

Exercise 20 What is $\partial_y \left[\strut A x^n y^m \right]$?

$A n x^{n-1} y^m$ $A n m x^{n-1} y^{m-1}$ $A m x^{n} y^{m-1}$ $A m y^{m-1}$

question id: drill-partialderivatives-20

Exercise 21 What is $\partial_{xy} \left[\strut A x^n y^m \right]$?

$A n x^{n-1} y^{m-1}$ $A n m x^{n-1} y^{m-1}$ $A m x^{n} y^{m-1}$ $A m x^{n-1} y^{m-1}$

question id: drill-partialderivatives-21

Exercise 22 What is $\partial_x \left[\strut f(x) + y\right]$?

$\partial_x f(x)$ $\partial_x f(x) + 1$ $\partial_x f(x) + y$ $0$

question id: drill-partialderivatives-22

Exercise 23 What is $\partial_x \left[\strut f(x) + g(y)\right]$?

$\partial_x f(x)$ $\partial_x f(x) + \partial_x g(y)$ $\partial_x f(x) + \partial_y g(y)$ $0$

question id: drill-partialderivatives-23

Exercise 24 What is $\partial_y \left[\strut f(x) + g(y)\right]$?

$\partial_x f(x)$ $\partial_x g(y)$ $\partial_y g(y)$ 0

question id: drill-partialderivatives-24

Exercise 25 What is $\partial_x \partial_y \left[\strut f(x) + g(y)\right]$?

$\partial_x f(x)$ $\partial_x \partial_y g(y)$ $\partial_y g(y)$ 0

question id: drill-partialderivatives-25

Exercise 26 What is $\partial_y \partial_y \left[\strut f(x) + g(y)\right]$?

Exercise 27 What is $\partial_y f(x) g(y)$?

$\partial_y g(y)$

$f(x)\ \partial_{y} g(y)$

$g(y)\ \partial_y f(x) + f(x) \ \partial_y g(y)$

question id: drill-partialderivatives-27

Exercise 28 What is $\partial_y h(x,y) g(y)$?

$\partial_y g(y)$ $g(y) \partial_y h(x, y)$ 0 $ g(y) _y h(x,y) + h(x,y) _y g(y)$

question id: drill-partialderivatives-28

Exercise 29 What is $\partial_x h(x,y) g(y)$?

$\partial_x h(x, y)$ $g(y)\ \partial_x h(x,y) + h(x,y)\ \partial_x g(y)$ $g(y) \partial_x h(x, y)$ $g(y) \partial_y h(x, y)$

question id: drill-partialderivatives-29

Exercise 30 What is $\partial_{yx} h(x,y) g(y)$?

$\partial_{yx} h(x, y)$

$g(y) \partial_{yx} h(x,y) + h(x,y)\ \partial_y g(y)$

$(\partial_y g(y)) \ (\partial_x h(x, y)) + g(y)\ (\partial_{yx} h(x, y))$

$(\partial_x g(y))\ (\partial_x h(x, y)) + g(y) (\partial_{xx} h(x, y) )$

question id: drill-partialderivatives-30

Exercise 31 What is the “with-respect-to” input in $\partial_y xy$?

$x$ $y$ $1$

question id: drill-partialderivatives-31

Exercise 32 What is the “with-respect-to” input in $\partial_x y$?

$x$ $y$ $1$

question id: drill-partialderivatives-32

Exercise 33 What is the “with-respect-to” input in $\partial_t y$?

$t$ $y$ $1$

question id: drill-partialderivatives-33

Exercise 34
At which of these inputs is the function steepest in the x-direction?

Exercise 35
At which of these inputs is the function practically flat?

Exercise 36
You are standing on the input point $(x=-1,y=4)$. In terms of the compass points (where north would be up and east to the right), which direction points most steeply uphill from where you are standing.

Exercise 37
You are standing on the input point $(x=2,y=1)$. In terms of the compass points (where north would be up and east to the right), which direction points most steeply uphill from where you are standing.

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