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20 Math Midterm 2 Review (Solutions)
Carolyn Stein Exam Date: November 9, 2011
Eigenvectors and Eigenvalues
What are Eigenvectors and Eigenvalues?
"Eigen" is German for "own" or "characteristic" If A~ = ~ , we say ~ is an eigenvector of matrix A with an associated eigenvalue v v v of If we view A as a transformation matrix, the eigenvectors are all vectors that are only scaled by the transformation Example: Find the eigenvectors and eigenvalues of the following matrices using geometric intuition: 5 0 Solution : Every vector is an eigenvector with eigenvalue of 5 0 5 0 1 Solution : Any vector on the line y = x is an eigenvector with 1 0 eigenvalue of 1. Any vector on the line y = x is an eigenvector with eigenvalue -1.
Finding Eigenvalues
A~ = ~ v v ) A~ v ~=0 v ) A~ v I~ = 0 v ) (A I)~ = 0 v We don't want ~ = 0, but we want (A v I)~ = 0. For this to be true, the v columns of (A I) must be linearly dependent, which implies: 1
det (A
a11 I) = det @ a21 a31
0
a12 a22 a32
a13 a23 a33
1
A=0
m1 m2 mn Solving this will yield an equation in the form ( ( ...( 1) 2) n) This equation is known as the characteristic polynomial 1 , 2 , ..., n are the eigenvalues Each eigenvalue has an associated algebraic multiplicitiy m1 , m2 , ...mn 1 2 Example: Find the eigenvalues and algebraic multiplicities of A = 4 3
Solution : -1, 5 both have algebraic multiplicity of 1
Finding Eigenvectors
For each eigenvalue , we can find the associated eigenvectors by solving: (A I)~ = 0 v
The space of solutions to the system is known as the eigenspace of The dimension of the eigenspace is known as the geometric multiplicity of 1 2 Example: Find the eigenspaces and geometric multiplicities of A = 4 3 1 1 Solution : 1 = t 5 = t both have geometric multiplicity 1 2 of 1
Eigenbases and Diagonalization
An eigenbasis is a basis made exclusively of linearly independent eigenvectors. Matrix A has an eigenbasis if the sum of the geometric multiplicities = n, the dimension of the matrix. If matrix A has a eigenbasis, it can be diagonalized. That is, it can be written in the form: 0 10 10 1 1 | ... | ... 0 | ... | 1 A = P DP 1 = @ v1 ... vn A @ ... ... ... A @ v1 ... vn A | ... | 0 ... | ... | n
Where P is a matrix composed of the eigenvectors, and D is a diagonal matrix with the corresponding eigenvalues.
2
Example: Does A =
If no, explain why not. 1 Solution : Yes - A = 1
1 4
2 3 1 2
have an eigenbasis? If yes, diagonalize it.
1 0
0 5
2 3 1 3
1 3 1 3
Functions of Multiple Variables
Level Sets
The level set of f corresponding to the value c is the collection of points {(x1 , x2 , ..., xn ) 2 Rn : f (x1 , x2 , ..., xn ) = c} If n = 2 the level set is also called a level curve If n = 3 the level set is also called a level surface Example: Sketch the level sets of f (x, y) = x2 + y 2 for c = {0, 1, 4, 9}. What does this function look like? Solution : level sets are concentric circles with increasing radius, the function is a cone around the positive z-axis Example: Sketch the level sets of f (x, y, z) = x2 + z4 for c = {0, 1, 4, 9}. How many dimensions does the function have? Can we visualize it? Solution : concentric elliptic cylinders around the y-axis with increasing radius. The function is 4D, so we can't really visualize it.
2
3
Common 3D Surfaces
sphere: x2 + y 2 + z 2 = a2 ellipsoid: cone:
x2 a2 x2 a2
+
y2 b2
+
z2 c2
=1
+
y2 b2
=
z2 c2
cylinder:
x2 a2
+
y2 b2
=1
x2 a2 y2 b2
elliptic paraboloid:
=
z c
Partial Derivatives
Limit Definition, Method
Let f (x, y) be a function of x, y The partial derivative of f with respect to x = lim Notation includes To calculate
@f @x , @f @x , x!0 f (x+ x,y) f (x,y) x
0 fx , and f1 - this review sheet intentionally uses all three
simply take the derivative of f (x, y) treating y as a constant
Examples: Find fx , fy , fxy and fyx for the following functions : xexy Solution: fx = exy + xyexy fxy = xexy + x(exy + yxexy ) = 2xexy + x2 yexy fy = x2 exy fyx = 2xexy + x2 yexy x2 sin y y 2 cos x
Solution: fx = 2x sin y + y 2 sin x fxy = 2x cos y + 2y sin x fy = x2 cos y 2y cos x fyx = 2x cos y + 2y sin x
4
Tangent Planes, Approximation
Let f (x, y) be a function of x, y and let (x0 , y0 ) be a point on the domain of f . The plane P (x, y) tangent to f at the point (x0 , y0 ) can be written as: P (x, y) = f (x0 , y0 ) + fx (x0 , y0 )(x x0 ) + fy (x0 , y0 )(y y0 )
P (x, y) = z0 + fx (x0 , y0 )( x) + fy (x0 , y0 )( y) If (x, y) is a point close to (x0 , y0 ), then we can use the tangent plane to approximate f (x, y) since P y) (x, f (x, y) Example: Write the equation of a tangent plane for f (x, y) = p point (2, 2) and use it to approximate 1.98 + 1.99 Solution : f (1.98, 1.99) 2 answer is 1.992486...)
1 4 (1.98
p
x + y at the
2)
1 4 (1.99
2) = 1.9925 (the exact
The Chain Rule
The General Chain Rule: Let f be a function f (x1 , x2 , ..., xn ) with each xi a function xi (t1 , t2 , ..., tm )
@f @tj
=
@f @x1 @x1 @tj
+
@f @x2 @x2 @tj
+ ... +
@f @xn @xn @tj
Picture and Intuition: The intuition is that tj influences each xi , which in turn influence f . Adding up all these effects should give the net effect on f . Example: Let f (x, y) = x2 + 2y and x = r sin t y = sin2 t. Find @f @t . Solution : @f @f @x @r = @x @r +
@f @t @f @y @f @y @y @r @y @t @f @r
and
= 2x sin t = 2r sin2 t = 2xr cos t + 2(2 sin tcost) = 2(r2 + 2) cos tsint
=
@f @x @x @t
+
Implicit Differentiation
If f (x, y) = c, we may want to calculate
@y @x
without solving for y.
In this case, we can implicity define y = y(x), take the derivative with respect @y to x on both sides, and solve for @x
5
@y Example: Find @x if x3 + y 3 = 6xy. (Hint: define f (x, y(x)) = x3 + y(x)3 6xy(x) = 0, then take partials of both sides)
Solution : @y 3x2 + 3y 2 @x
@y (6y + 6x @x ) = 0 )
@y @x
=
6y 3x2 3y 2 6x @xi @xj
More generally, if f (x1 , x2 , ..., xn ) = c, then Proof:
=
fx j fx i
@ @ (f (x1 , x2 , ..., xn )) = (c) @xj @xj @f @xi @f + = 0 (Why is this true?) @xi @xj @xj fxj @xi @f @f = / = @xj @xj @xi fxi
Hessian Matrix, Young's Theorem
The Hessian Matrix is a matrix of second-order partial derivatives: 0
00 f11 00 B f21 B @ ... 00 fn1 00 f12 00 f22 ... 00 fn2
... ... ... ...
1 00 f1n 00 f2n C C ... A 00 fnn
Young's Theorem:
00 00 fij = fji 000 000 000 000 000 000 fijk = fikj = fjik = fjki = fkij = fkji
In other words, the order of differentiation doesn't matter, as long as all the partial derivatives are continuous.
Note that this implies that the Hessian is symmetric.
Gradient and Directional Derivative
The Gradient Vector
The gradient vector is a vector of a function's partial derivatives. rf = hfx1 , fx2 , ..., fxn i 6
The Directional Derivative
The directional derivative of f (x, y) in the direction of ~ (where ~ = hu1 , u2 i u u is a unit vector) is the scalar quantity: D~ f = ~ rf = u1 fx + u2 fy u u What is the interpretation of this? fx is the slope of f in the x-direction, fy is the slope of f in the y-direction, and so D~ f is simply the slope of f in the u ~ -direction. u Example: Find the directional derivative of f (x, y) = x2 y 3 (-2,1) in the direction of ~ = h2, 5i. v Solution :
32 p 29
4y at the point
Properties
1. rf (x, y) is orthogonal to the level curve f (x, y) = c 2. rf (x, y) points in the direction of maximal increase of f (x, y) 3. The slope of maximal increase is |rf (x, y)|
Unconstrained Maximization / Minimization
Finding Stationary Points
To find the stationary points or critical points, set fx = 0, fy = 0 and find all x, y that satisfy the system of equations. A stationary point can be a local maximum, local minimum, or saddle point. Note that this can be extended to more variables - simply set every partial equal to 0 and solve the resulting system. Example: Find the stationary points of f (x, y) = x2 + y 2 + y Solution : fx = 2x = 0 ) x = 0 fy = 2y + 1 = 0 ) y = 1 2 (0, 1 ) is the only stationary point 2 1.
7
The Extreme-Value Theorem, Classifying Stationary Points
The Extreme-Value Theorem states: If f is a continuous function on a closed and bounded set S 2 Rn , there exists a = ha1 , a2 , ..., an i 2 S and b = hb1 , b2 , ..., bn i 2 S such that f (b) f (x) (a) for any x 2 S. In other words, an absolute maximum and absolute minimum always exist for a continuous function on a closed and bounded set. You can use the Extreme-Value Theorem to find the absolute maximums and minimums of f on S: 1. Find all stationary points of f that lie inside S 2. Find the smallest and largest values of f that lie on the boundary of S 3. These are all the possible candidates for extreme values - plug these points into f to determine which is the absolute maximum and minimum Example: Find the extreme points and extreme values of f (x, y) = x2 + y 2 + y 1 that lie inside S = {(x, y) : x2 + y 2 1}. Solution : The stationary point found (0, 1 ) above is in S. Now plug in 1 for x2 + y 2 2 since we want to examine the boundary. f (x, y) = 1 + y 1 = y. This is clearly maximized/minimized at (0,1) and (0,-1). If we plug all 3 points into f we see that (0,1) is the absolute maximum and (0, 1 ) is the absolute minimum. 2
8

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MATH 1A: OPTIMIZATION (PART II)1. The Statue of Liberty is 32 m tall, and rests on a base 49 m tall. How far away from the base should you stand to get the best view of the statue? (In other words, to maximize the angle subtended by the statue at your ey

Harvard - MATH - 1a

^ L'HOPITAL'S RULE1. Find lime3t - 1 . t0 t2. Find limln x . x1 sin x3. What is limsin2 (x) ? x0 x2 - x34. Last night, I was trying to compute limx0 my conclusion. What is going on?sin(3x) x2 +x .The numerical evidence does not seem to agree with

Harvard - MATH - 1a

MATH 1A: ANTIDERIVATIVES1. Water is leaking out of a tank at the rate of (10 - t) liters per minute, where t is time in minutes. The leaking starts at t = 0. How much water has leaked after 4 minutes? What if the leak had started at t = 1?2. For each of

Harvard - MATH - 1a

AREAS AND DISTANCES1. Express the following sums using the notation: (1) 13 + 23 + + 1003(2) 1 + 3 + 5 + + 101(3) 1 -1 2+1 3-1 4 +1 292. Write down the definition of the area under the given function on the given interval. Approximate the area

Harvard - MATH - 1a

MATH 1A: THE DEFINITE INTEGRAL31. Approximate the definite integral than or greater than the actual value?0(2 - x)dx using six rectangles and right endpoints. Is this value lessCan you find the exact value of the integral? (Hint: What does the graph

Harvard - MATH - 1a

EVALUATING DEFINITE INTEGRALS1. Evaluate the following integrals.2(1)0(6x2 - 4x + 5)dx2(2)cos d1(3)04 dx 1 + x2/4(4)01 + cos2 cos2 d2. What is wrong with the following equation?3 -11 1 dx = - x2 x3 -14 =- . 3Date: November 23, 2009.

Harvard - MATH - 1a

MATH 1A: THE FUNDAMENTAL THEOREM OF CALCULUS1. Let f (t) be the function defined on the interval [-5, 5] by the graph shown. Define the area function F by f x4 2xF (x) =0f (t) dt.531 2135x(1) Where is F increasing and decreasing?(2) Where i