Week 20 - Probabilistic Models, Diusion and Genetics
Section 6.2
TEST PREPARATION PROBLEMS
1) The discrete-time dynamical system is pt+1 = 0.7pt because the probability it remains is
0.7 each second. The solution is pt = 0.7t . After 10 seconds, it remain
Week 17 - Tangent Planes, Functions of Many Variables, and Vectors in R2
and R3
Hughes-Hallett Section 12.5
SUGGESTED PROBLEMS
In Exercises 3-5, represent the surface whose equation is given as the graph of a two-variable
function, f (x, y), and as the le
Week 19 - Higher Partial Derivatives, and Optimization
Hughes-Hallett Section 14.7
SUGGESTED PROBLEMS
In Exercises 1-10, calculate all four second-order partial derivatives and check that fxy = fyx .
Assume the variables are restricted to a domain on whic
Week 17 - Tangent Planes, Functions of Many Variables, and Vectors in R2
and R3
Hughes-Hallett Section 13.1
SUGGESTED PROBLEMS
For Exercises 9-14, perform the indicated operations on the following vectors:
9. 5b
5b = 15i + 25j + 20k
10. a + z
a + z = (2k
Week 19: Higher Partial Derivatives, and Optimization
Hughes-Hallett Section 14.7
SUGGESTED PROBLEMS
Find the quadratic Raylor Polynomials about (0, 0) for the functions in Exercises 11-18.
11. f (x, y) = (x y + 1)2
f = (x y + 1)2
fx = 2(x y + 1)(1) = 2(x
Week 19 - Higher Partial Derivatives, and Optimization
Hughes-Hallett Section 15.1
SUGGESTED PROBLEMS
1. Which of the points A, B, C in Figure 15.17 appear to be critical points? Classify those
that are critical points.
Figure 15.17
A is not a critical po
Week 18 - Directional Derivatives and the Gradient; the Chain Rule
Hughes-Hallett Section 14.6
SUGGESTED PROBLEMS
For Exercises 1-6, nd dz using the chain rule. Assume the variables are restricted to domains
dt
on which the functions are dened.
1. z = xy
Week 18 - Directional Derivatives and the Gradient; the Chain Rule
Hughes-Hallett Section 14.4
SUGGESTED PROBLEMS
In Exercises 15-28 nd the gradient of the given function. Assume the variables are restricted
to a domain on which the function is dened.
4
3
Week 17 - Tangent Planes, Functions of Many Variables, and Vectors in R2
and R3
Hughes-Hallett Section 14.3
SUGGESTED PROBLEMS
For the functions in Exercises 1-4, nd the equation of the tangent plane at the given point.
1
1. z = (x2 + 4y 2 ) at the point
Week 18 - Directional Derivatives and the Gradient; the Chain Rule
Hughes-Hallett Section 14.5
SUGGESTED PROBLEMS
In Exercises 1-10, nd the gradient of the given function.
5. f (x, y, z) = xey + ln(xz)
fx = ey +
1
z
xz
fy = xey
1
x
fz =
xz
So grad f = f =
Week 19 - Higher Partial Derivatives, and Optimization
Hughes-Hallett Section 15.2
TEST PREPARATION PROBLEMS
Do the functions in Exercises 3-7 have global maxima and minima?
3. f (x, y ) = x2 2y 2
The function doesnt have a global max, because f + as x .
Week 20 - Probabilistic Models, Diusion and Genetics
Section 6.1
TEST PREPARATION PROBLEMS
17)
19)
21)
MATH 122 - Section 6.1 Solutions
2
23)
28)
This island can be described by a simple deterministic rule; occupied for 2 years and
extinct for 1 year. It