Consider the matrix A given by [ 2 0 0 ] [ 0 2 3 ] [ 0 3 10 ] (20) Find all its eigenvalues and corresponding eigenvectors. Show your work. (+5) Write down the entire eigendecomposition (i.e. the matrices X, Lambda, and X inverse) explicitly.
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Taylor Polynomial HW
1) Evaluate cos ( 2 π / 3 ) on your calculator and using the
first 4 terms of the TP for cos x.
2) Integrate cos ( x^3 ), from 0 to π / 6, using the first 3 terms
of the TP for cos x.
3) Evaluate e^x at x = .4 on your calculator and using the first 5
terms of the TP for e^ x.
4) Integrate e^x3, from 0 to .3, using the first 3 terms of the TP
for ex.
5) If I integrate 1/(1-x) I will get - ln (1 - x). Integrate the
given TP for 1/(1-x). What is the TP for - ln ( 1 - x )?
6) What is the value of, - ln ( 1 - x ) if x = .3? I got .3567. Use
the first 4 terms of the TP you created in question 5 and see if
you obtain the same result. I got .3560.
Here are other Taylor Polynomials for other trig functions:
tan ( x ) = x + (1/3) x3 + (2/15) x5 + (17/315) x7 + (62/2835) x9 +
....
sec ( x ) = 1 + (1/2) x2 + (5/24) x4 + (61/720) x6 + ...
7) Find the integral of tan x from 0 to π / 6. Use the first 3
terms of the TP.
8) Find a TP for sec 2 x, recall sec 2 x is the derivative of tan
x.
9) On your calculator, what is the cos (π / 3)? You should get 1/2.
Obviously, the sec (π / 3) is 2. Use the first 4 terms of the TP
for sec x and see if the answers agree.
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Find the flux of the vector field
V(x, y, z) = 7xy2i + 2x2yj + z3k
out of the unit sphere.
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3. Determine whether the following properties can be satisfied by a function that is continuous on the interval (−∞, ∞). If such a function is possible provide an example or a sketch of the function. If such a function is not possible explain why.
a. A function f is concave down and positive everywhere.
b. A function f is increasing and concave down everywhere.
c. A function f has exactly 2 local extreme and 3 inflection points.
d. A function f has exactly 4 zeros (x-intercepts) and 2 local extreme.
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identify the types of symmetry present in the polar functions.
r = 2 + 3 sin (theta)
r = 5 cos 2 (theta)
r^2 = 9 sin 2 (theta)
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Use the techniques of Chapter 4 to sketch the graph of y= x^4/4+x^3/3-x^2.
a) domain,
b) y-intercept,
c) asymptote(s),
d) intervals of increase and/or decrease,
e) local maximum(s) and/or local minimum(s),
f) intervals of concavity,
g) points of inflection. (For full credit, remember to show all work and include sign charts for the Increasing/Decreasing and 2nd Derivative tests.)
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Consider the function f(x)f(x) whose second derivative is f''(x)=5x+10sin(x)f′′(x)=5x+10sin(x). If f(0)=4f(0)=4 and f'(0)=4f′(0)=4, what is f(5)f(5)?. show work
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Find the equation (in terms of x and y) of the tangent line to the curve r=5sin5θ at θ=π/3.
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Given the function f(?) = ?3 − 9?2 + 24? − 2
a) Find the critical numbers and make a sign diagram for the first derivative.
b) Find the possible inflection points and make a sign diagram for the second derivative.
c) Using the information to sketch the graph of the function and show the local mins and maximums and the inflection points on the graph.
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Find y as a function of x if
y′′′+25y′=0
y(0)=2, y′(0)=20, y′′(0)=−100
y(x)=
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Given the line integral ∫c F(r) · dr where
F(x, y, z) = [mxy − z3 ,(m − 2)x2 ,(1 − m)xz2 ]
(a) Find m such that the line integral is path independent;
(b) Find a scalar function f such that F = grad f;
(c) Find the work done in moving a particle from A : (1, 2, −3) to B : (1, −4, 2).
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Show all work for credit. Be sure to label work and use sentences as appropriate to explain what steps you are taking. Be sure to clearly label any steps, results, or conclusions. Mention every single step you do.
For the following function f(x) = 2x-5/x+1 Find :
(a) (5 points) Find the domain (in interval notation).
(b) (5 points)Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote or hole.
(c) (5 points)Find any Horizontal Asymptotes. Show work or write a sentence to substantiate your caim.
(d) (5 points)Find all x and y intercepts
(e) (10 points)Graph, finding additional points as needed.
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Describe the differences, benefits and weaknesses of manual AIS systems, legacy AIS systems and modern, integrated AIS Systems.
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5. Solve the nonlinear inequality. Express the solution using interval notation.
x + 3 > −2x−5
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