Free Printable MTH231

1.

If $g\left(x,y\right)=xe^{xy^2}$ then $g_x\left(x,y\right)$ is

a)

$g_x\left(x,y\right)=e^{xy^2}\left(1+x^2y\right)$

b)

$g_x\left(x,y\right)=e^{xy^2}\left(1+xy^2\right)$

c)

$g_x\left(x,y\right)=xy^2e^{xy^2}$

d)

$g_x\left(x,y\right)=xe^{xy^2}$

e)

$g_x\left(x,y\right)=2x^2ye^{xy^2}$

2.

What is the domain of the function $z=\sqrt{\ 9-x^2-y^2}$

a)

The domain is all (x,y) such that $x^2+y^2=9$

b)

The domain is the set of all points $\left(x,y\right)$ for which $x^2+y^2\ge0$

c)

The domain is the circular disk of radius 3 with center at the origin

d)

The domain is the set of all points $\left(x,y\right)$ for which $x^2+y^2\ge9$

e)

NOTA

3.

If $f\left(x,y\right)=xy^2+x^3$ then $f_x\left(2,-1\right)\$ is

a)

10

b)

7

c)

5

d)

13

e)

8

4.

Evaluate the $\lim_{\left(x,y\right)\rightarrow\left(1,1\right)}\frac{2x^2-xy-y^2}{x^2-y^2}$

a)

1.5

b)

0

c)

1

d)

DNE

e)

2

5.

Evaluate $\lim_{\left(x,y,z\right)\rightarrow\left(-1,0,4\right)}\frac{x^3-ze^{2y}}{6x+2y-3z}$

a)

3/5

b)

NOTA

c)

2/3

d)

5/18

e)

1/6

6.

Let $f:\left[a,b\right]\rightarrow R$ be continuous. Suppose that $f\left(a\right)<f\left(b\right)$ . Then for any $u$ with $f\left(a\right)<u<f\left(b\right)$ there exists a $k\ \in\ \left(a,b\right)$ such that ?

a)

f(k) > f(b) for some k in (a, b)

b)

f(k) = f(a) for some k in (a, b)

c)

f(k) = u for some k in (a, b)

d)

f(k) = u for some k not in (a, b)

e)

f(k) < u for some k in (a, b)

7.

If $f\left(x,y\right)=x^3e^{5y}+y\sin\ 2x$ then $f_{xx}$ is

a)

$6xe^{5y}-4y\sin\ 2x$

b)

$6xe^{5y}+4y\cos\ 2x$

c)

$6yxe^{5y}+4x\sin\ 2x$

d)

$3x^2e^{5y}+2y\cos\ 2x$

e)

NOTA

8.

Approximate the change in $z=xy^2$ from its value at (0.5, 1.0) to its

value at (0.503, 1.004).

a)

0.007

b)

0.005

c)

0.012

d)

NOTA

e)

0.003

9.

Determine the formula for $\frac{\partial w}{\partial t}$ given that $w=w(x,y)\ \ \ x=x(p,q,s)\ \ \ y=y(p,u,v)\ \ s=s(u,v)\ \ \ p=p(t)$

a)

$\frac{\partial w}{\partial t}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial p}\frac{\partial p}{\partial t}$

b)

$\frac{\partial w}{\partial t}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial s}\frac{\partial s}{\partial t}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial p}\frac{\partial p}{\partial t}$

c)

$\frac{\partial w}{\partial t}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial p}\frac{\text{d}p}{\text{d}t}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial p}\frac{\text{d}p}{\text{d}t}$

d)

$\frac{\partial w}{\partial t}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial q}\frac{\partial q}{\partial u}\frac{\partial u}{\partial t}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial v}\frac{\partial v}{\partial s}\frac{\partial s}{\partial t}$

10.

Compute $\frac{\text{dy}}{\text{d}x}$ for the equation $x^2y^4-3=\sin\ \left(xy\right)$ at $\left(-\frac{7}{5},0\right)$

a)

7/5

b)

-3

c)

1

d)

-1

e)

NOTA

MTH231

Total questions: 10

Worksheet time: 13mins

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