Math 9 Practice Final

1.

(a)

Find the Taylor series about x=0 of

$$\arctan\left(\frac{x}{2}\right).$$

(b)

What is the radius of convergence of this series?

2.

(a)

Find an equation for the tangent plane to the graph of f(x,y) = xy + y² + e^2x at (0,2,5).

(b)

Find an equation for plane parallel to this tangent plane containing the point (1,1,0).

(c)

Where does the line perpendicular to this new plane and containing (1,1,0) hit the original tangent plane.

3.

Let f(x,y) = xy and $g(x,y) = e^y \sin(x)$. Describe the points where the level sets of f and g are perpendicular.

4.

Find the minimum and maximum values of f(x,y) = (x + y)e^{-(x2 + y2)} on the disk $x^2 + y^2 \leq 1$.

5.

Let $f(x,y,z) = ((yz) \sin(xy) , (yz)e^{(xz)} + (xy))$.

Note: If you have already tried this the point was meant to be

$$(\frac{\pi}{2},0 ,0)$$

not

$$\[ (\frac{\pi}{2},1,1)$$

as stated. Both ways are doable but this second point is much easier.

(a)

Express f as a composite of a function $g: R^3 \rightarrow R^2$ and a function $h: R^3 \rightarrow R^3$.

(b)

Find Dh and Dg (Df refers to the derivative of f).

(c)

Use the chain rule to compute $Df(\frac{\pi}{2},1,1)$.

(d)

Find the kernel of $Df(\frac{\pi}{2},1,1)$.

6.

Notice

$$M_{\lambda} = \left[\begin{array}{rr} \lambda & 0 \\ 0 & 1 \end{array} \right]$$

is a stretch of the plane by $\lambda$ in the direction of the x-axis leaving the orthoganal direction fixed. Let $M^{\theta}_{\lambda}$ denote the stretch by $\lambda$ in the direction determined by the line inclined at an angle $\theta$ with respect to the x-axis with its orthogonal direction fixed. Find a formula for $M_{\lambda}^{\theta}$.

7.

Let V be the vector space of polynomials of degree $\leq 3$ and let A denote the linear transformation of V to V given by

$$A(p(x)) = a \frac{d^2p}{dx^2} + b \frac{dp }{dx} + c p .$$

(a)

Express A as a matrix.

(b)

In each of the following cases describe the rank of A, and find a non-zero polynomial in A's kernel whenever such an element exists.

i.

$a \neq 0$ and $b \neq 0$ and $c \neq 0$.

ii.

$a \neq 0$ and $b \neq 0$ and c = 0.

iii.

$a \neq 0$ and b = 0 and c = 0.

8.

(a)

Find the general solution to

$$\frac{d^2f}{dx^2} - 3 \frac{df }{dx} + 2 f= 0 .$$

(b)

Use the previous problem (problem 7(a)) to find a polynomial solution to

$$\frac{d^2f}{dx^2} - 3 \frac{df }{dx} + 2 f = x^3 + 3x +1.$$

(c)

Find a solution to

$$\frac{d^2f}{dx^2} - 3 \frac{df }{dx} + 2 f = x^3 + 3x +1$$

satisfying f(0)=1 and $\frac{df}{dx}(0) = 2$. Note: you are the right track if you get coeffeicients wich are rational numbers with denominators like 8.