MA 5320,Advanced calculus, Midterm II, Fall 2000

Show all your work.

1.

Find the following limit.

(a)

$${\lim_{x\rightarrow 0} \frac{x^2 - \sin ^2 x}{x^4}}$$

(b)

$${\lim_{x\rightarrow 0^+} \frac{\ln \cos 4x}{\ln \cos 2 x}}$$

(c)

$${\lim_{x\rightarrow 0} \frac{x + \sin x}{2x+ 1}}$$

2.

Determine if f(x,y) is continuous at (0,0) where

$$f(x,y)= \left\{ \begin{array}{ll} \frac{xy}{x^2 + y^2} ~~~&(x,y) \ne (0,0) \\ 0 & \mbox{otherwise.} \end{array}$$

3.

Do one of the two.

(a)

Show that $U(x,y,z)=\frac{1}{\sqrt{x^2 + y^2 +z^2}}$ satisfies laplace's partial equation

$$\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2}+ \frac{\partial^2 U}{\partial z^2}=0$$

(b)

Show that any continuous map on the closed interval [0,1] is uniformly continuous. Hint: Use the compactness of [0,1.

4.

Use $\epsilon-\delta$ method. Prove that f(x) = x² - 1 is continous at x=3.

5.

What is the difference between being uniformly continuous and continuous.

6.

Find extrema value of 4xy subject to x+y = 8.

7.

Find the relative maxima and relative minima of

f(x,y) = x³ + y³ -12 x - 3 y +20

8.

(a)

Find the Jacobian determinant of f, g where f(x,y)=x² +2 xy, g(x,y)=e^x - xy².

(b)

f(x,y,z)=4 x² + 7 y² + 3z². at (1,1,1) that has the largest directional derivate.

(c)

Find the derivative of f with respect to t where $f(x,y,z)=x^2 +y^2 +z^2, x = \sin t, y=\cos t, z=t$.

9.

(a)

Find the tangent plane to the surface xy²z + 3x²=2yz² - 8 z at the point (2,1,-1).

(b)

Find the tangent line to the curve

$$x = \cos t, y= \sin 3 t, z = \cos t$$

at the point where $t = \frac{\pi}{4}$.

10.

z=f(x,y) =x² y - 3 y.

(a)

Find dz.

(b)

Find the Taylor expension of f(x,y) at (x,y)=(2,1) upto the second order.

11.

A region $\Omega$ in xy plane is bounded by x + y = 8, x+ y = 4, x- y = 0 and x - y = 4.

(a)

Determine the region $\Omega '$ in the uv planne into which $\Omega$ is mapped under the transformation u=x+y, and v=x-y.

(b)

Find the jacobian determinant of of x and y with respect to u and v.

(c)

Find the area of $\Omega, \Omega '$.

12.

(a)

f(0)= -2 , f(3)=3. To conclude that there is a root in (0, 3), what condition does f have to satisfy?

(b)

f(0)= -2 , f(3)=-2. To conclude that there is a critical point in (0, 3), what condition does f have to satisfy?

13.

Determine if the following statement is true or false.Give a brief reasoing or quote theorem to supposrt your answer.

(a)

Image of an open set under continuous map is an open set.

(b)

Inverse image of a compact set under a continuous map is always compact.

(c)

Inverse image of a compact set under a continuous map is always compact.

(d)

Image of a closed rectangle in R² under a continuous map is a parallegram.

(e)

Image of a closed rectangle in R² under a continuous map is convex.

(f)

Image of a closed rectangle in R² under a continuous map is connected.

(g)

A Continous map on a compact set $\Omega$ always assumes maximam and minimum value in $\Omega$.

(h)

f(10)=0, f(4) =0. f is differentiable in [3,11]. Then there is always a critical point in (4,10).

(i)

Every polynomial in Rⁿ is continuous and differentiable everywhere.

(j)

Every continous map is uniformly continuous.

(k)

Every rational map in R is continuous.