Assignment 0

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course MTH 279

6/16 2300

Question: `q001. Find the first and second derivatives of the following functions:

3 sin(4 t + 2)

2 cos^2(3 t - 1)

A sin(omega * t + phi)

3 e^(t^2 - 1)

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Your solution:

d/dx 3 sin(4 t + 2) = 12cos(4t+2)

d2t/dx2 3 sin(4 t + 2) = -48sin(4t+2)

d/dx 2 cos^2(3 t - 1) = -12sin(3t-1)

d2t/dx2 2 cos^2(3 t - 1) = -36cos(3t-1)

d/dx A sin(omega * t + phi) = A*omega*cos(omega*t+phi)

d2t/dx2 A sin(omega * t + phi) = -A*omega^2*sin(omega*t+phi)

d/dx 3 e^(t^2 - 1) = 6te^(t^2-1)

d2t/dx2 = 3 e^(t^2 - 1) = 12t^2e^t^2-1)

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Your derivatives are correct on two of the functions. At least one of the derivatives is incorrect on each of the other two.

You should revise and include the steps you used in taking the derivatives.

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Question:

`q002. Sketch a graph of the function y = 3 sin(4 t + 2). Don't use a graphing calculator, use what you know about graphing. Make your best

attempt, and describe both your thinking and your graph.

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Your solution:

The graph of sin(t) is manuipulated by the constant multiplyer 3, making the new graph 3-times as high as the normal sin(t) plot. The 4t+2 expression on the interior of the function causes the sin curve to be at a y-value of zero

when 4t+2 is equal to zero, which is at t=-1/2. The resulting graph is a sin curve which starts at t=-1/2 and moves in a positive direction, attaining a maxinmum value at y=3 and a minumum value of y=-3.

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Question:

`q003. Describe, in terms of A, omega and theta_0, the characteristics of the graph of y = A cos(omega * t + theta_0) + k.

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Your solution:

The above defined graph has the folowing characteristics.

- It is a cos curve with a phase shift equal to the t value when (omega*t+theta_0)=0

- The maximum value of the graph is at A, and the minimum is at -A

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Right, but you should have solved for the phase shift, which is -theta_0 / omega.

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Question:

`q004. Find the indefinite integral of each of the following:

f(t) = e^(-3 t)

x(t) = 2 sin( 4 pi t + pi/4)

y(t) = 1 / (3 x + 2)

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Your solution:

int f(t) = (-e^(-3t))/3 + C

int x(t) = (-cos(4*pi*t+pi/4))/4*pi + C

int y(t) = ln(3x+2)/3 + C

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You should show your steps, but these are correct.

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Question:

`q005. Find an antiderivative of each of the following, subject to the given conditions:

f(t) = e^(-3 t), subject to the condition that when t = 0 the value of the antiderivative is 2.

x(t) = 2 sin( 4 pi t + pi/4), subject to the condition that when t = 1/8 the value of the antiderivative is 2 pi.

y(t) = 1 / (3 t + 2), subject to the condition that the limiting value of the antiderivative, as t approaches infinity, is -1.

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Your solution:

F(t) = -e^(-3t)/3 - 1/3

X(t) = the indefinite integral is (-cos(4*pi*t + pi/4))/ (2*pi) + C

I dont know how to simply solve for the constant of integration

Y(t) = the indefinite integral is (ln(3t+2))/t

I dont know what ""limiting value of the antiderivative, as t approaches infinity, is -1."" means.

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For this function when t = 0, F(t) = 0, not 2.

Your general solution is

F(t) = -e^(-3t) / 3 + C.

F(0) = 2.

Substitute to get the expression for F(0), set equal to 2 and find C.

The same strategy works for the others.

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Question:

`q006. Use partial fractions to express (2 t + 4) / ( (t - 3) ( t + 1) ) in the form A / (t - 3) + B / (t + 1).

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Your solution:

A = 5/2

B = -1/2

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Your solution does not show how you used partial fractions to get these results.

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Question:

`q007. The graph of a function f(x) contains the point (2, 5). So the value of f(2) is 5.

At the point (2, 5) the slope of the tangent line to the graph is .5.

What is your best estimate, based on only this information, of the value of f(2.4)?

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Your solution:

F(2.4) ~ 5.8

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This isn't correct, but in any case you need to show the steps of your reasoning.

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Question:

`q008. The graph of a function g(t) contains the points (3, 4), (3.2, 4.4) and (3.4, 4.5). What is your best estimate of the value of g ' (3), where the ' represents the derivative with respect to t?

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Your solution:

g'(3) ~ 1/2

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This is the slope between the first two points, but there is a third point, which indicates a trend to the slopes. So I would call this an almost-adequate estimate, but it wouldn't qualify as a good attempt at a best estimate.

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You're good on the trigonometric graphs and some of the basic calculus, and I suspect on some other topics.

You've also got some clear errors, and I want to really see what you know about these questions, so I'll ask for a revision.

You don't want any of this stuff getting in your way when you're trying to learn this subject.

&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).

Be sure to include the entire document, including my notes.

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