Annotated Math HW

8/11/2019 Annotated Math HW

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Helpful Questions To Get Yourself Thinking

What about the question/dirrections helped you identify the type of problem?

What were you thinking as you solved this question?

Is there a general approach to this type of problem?

Is there a way to break solving this type of problem down into steps/simpler parts?

Are there theorems/formulas important to this problem?

Why did you do what you did?

How did you know to do what you did (at each step)?

How did you know what your next step should be?

What is you ultimate goal as you solve this problem? (For example, what format were you

working towards as you simplified an expression?) What would you have done differently with a different problem of this type? (For example,

is there something you often do with other questions of this type, but didnt have to do withthis particular question?)

Can you compare/contrast this question with other questions?

How would you explain your solution to a classmate?

What questions where you asking yourself as you solved this problem?

What questions/comments did Jackie make about this?

What questions (or hints) would an instructor in the Calc Help Room ask you as you solvedthis problem?

(If the question involves some guess and check) Why did you make the guesses you did?

Are there any relevant vocabulary words you should be using?

If you used a theorem, did you explain how all the hypothesis were met?

These questions are only provided as guidelines to get you thinking (and may not apply to everyproblem). My hope is that as the semester goes on and you are more used to writing the annotatedproblems, you will have an easier time with the thought process behind the math problems, andyou wont need to rely on these questions as much.


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Rubric

Please note that every problem will be different, and not all of the criteria will apply to every problem, so the Rubric will be slightlydifferent for each question. Additionally, your solution may not fall exactly into one of these categories, so this Rubric is only meantas a general guide.

0 Annotated problem didnt follow one of the directions, isnt complete, or isnt mathematically correct.1 Problem is mathematically correct and all work is shown.

Annotation are missing important information, are trivial, or are incorrect.2 Problem is mathematically correct, but not enough of the work is shown.

Annotations are adequate (meet the criteria for a 3).Problem is mathematically correct and all work is shown.Annotations ignore or inadequately explain one important step in the problem.

3 Annotations are otherwise clear, correct, and well written, and demonstrate understanding of relevant mathematical conceptsAnnotations make at least some big picture connections, connections to other problems of that typeor connections to other calculus concepts.Problem is mathematically correct and all work is shown.

4 Annotations explain all important steps in the problem, including set up.Annotations meet all other criteria for a 5 except one.Problem is mathematically correct and all work is shown.Annotations explain all important steps in the problem, including set up.

5 Annotations are clear, correct, and well written.Annotations are insightful and demonstrate a real understanding of the mathematical concepts behind the solution.

Annotations make appropriate big picture connections to other problems of that type and to other calculus concepts.

Its more of an art than a science to determine how much of the calculations/simplifying you need to include annotations for.One guideline that might help you is if you can describe several lines of the calculations as a process you learned in Calc 1 (or earlierthis semester), feel free to just reference that. Even better is to indicate that the calculations are an already learned process, followedby an additional comment as relevant.

You dont need to indicate each algebra step in simplifications (unless there is some very clever trick in the process). You may juststate something along the lines of Algebra Simplifications.


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Sample Annotated Problem Problem Set 0 Question 1

1. Find the derivative of the following functionsa. f(x) = (17x2 13)(x 1)

b. y= (3x cos x)5

When taking a derivative, the best case scenario is if we can use the derivative formulasfrom section 3.3 and take the derivative by inspection. (Constant Multiple Rule,Sum/Difference Rule, Power Rule, Derivative of a Constant Function.) If this isimpossible we can simplify the function using algebra, or use another derivative rule,such as the Product, Quotient, or Chain Rule to begin to take the derivative, hopefullygetting down to something simpler where we can use the simpler rules.

Part a)

f(x) = (17x2 13)(x 1)

f(x) = 17x3 13x 17x2 + 13

f(x) = 317x2 13 217x+ 0

The right side isnt simple enough to take the

derivative by inspection. We can either simplify(multiply the polynomials) or use the productrule. Either would work here, since we need tochoose one, we will simplify.

Now we have a polynomial, so we can takethe derivative by inspection

Whats going on behind the derivative short cut:

(Take the derivative of both sides)

ddx

f(x) = ddx

(17x3 13x 17x2 + 13)

(Use the Sum Rule, since polynomials are differentiable,then use the fact that f(x) is just notation for d

dxf(x) )

f(x) = d

dx17x3

d

dx13x

d

dx17x2 +

d

dx13

(Use Constant Multiple Rule, since polynomials are diff.)

f(x) = 17 d

dx

x3 13 d

dx

x 17 d

dx

x2 + d

dx

13

(Use Power Rule [ ddx

(xn) = nxn1 for n a real number],and Deriv. of Constant Function)

f(x) = 17(3x2) 13(1x0) 17(2x1) + 0

(Simplify with algebra)

f(x) = 317x2 13 217x+ 0


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Part b)

Looking at this function we see immediately that this isnt something we can easily simplifyinto a polynomial so we can evaluate it by inspection, like in part a). Our next course ofaction is to look at the structure, and determine if any of the Product, Quotient or Chainrules might help us. In this case we see that this function has the format of 2 functionsthat are composed with each other. (Its a big mess of stuff, all raised to the 5th power).Thats our clue that we should start with the Chain Rule.

y = (3x cosx)5

dy

dx = 5(3x cos x)4

d

dx(3x cosx)

dy

dx = 5(3x cos x)4 (3 + sin x)

Using the shortcut, the trick is to identifywhat the outermost thing going on is. We seethat the outermost thing here is the 5th power.Once weve done this were ready to write ourfirst step of work with the shortcut method.

However, to show that math behind this shortcut, we decompose y, and write it as y = u5

whereu= 3x cosx). Then, the chain rule for-

mula says dy

dx = dy

dududx . For this to be useful, we

calculate the derivative of y with respect to u(thinking of u as the variable). Since y = u5,dy

du = 5u4. Now we find the derivative of u

with respect to x, since u = 3x cosx, we seedudx

= 3 sin x.Now we begin to substitute into the chain rule

formula in an order that will help it match thescratch work that were accustomed to showing.

dydx

= dydu

dudx

= (5u4)du

dx

= 5(3x cos x)4du

dx

= 5(3x cos x)4 d

dxu

= 5(3x cos x)4 d

dx(3x cos x)

(now we finish substituting and simplifying)

= 5(3x cos x)4(3 sinx)

= 5(3x cos x)4 (3 + sin x)

We also notice that we could have used function notation, instead of the Leibniz notationwe used. And finally that the full solution in the annotations could have been shorter ifwe werent trying to match the scratch work were accustomed to showing.


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Sample Annotated Problem Problem Set 0 Question 2

Find the absolute maximum and absolute minimum values offon the given interval.f(x) = 4

3x3 2x2 + 3, [2, 3]

Since [

2, 3] is a closed interval (indicated by the square brackets),and since we were asked to find an absolute max/min, we want to use theClosed Interval Methodfor this problem.

We can only use the Closed Interval Methodif the function is continuous onthe interval. Since f(x) =x4 2x2 + 3 is a polynomial it is continuouseverywhere, and in particular, it is continuous on [2, 3],so we can use the Closed Interval Method.

1. Find the values offat the critical numbers off in (2, 3)a) Find critical numbers off(x),

first find f(x), then set f(x) = 0 and solve for x.b) find the f values

plug the critical numbers into f(x) (unless they are outside (2, 3)).

Crit Numbersf(x) = 4

3x3 2x2 + 3

f(x) = 4x2 4x find f(x), see Chapter 30 = 4x2 4x set f(x) = 00 = 4x(x 1) solve for x, by factoring

x= 0 or x= 1 are the critical numbers, candidates for local maxsince both 0 and 1 are in the interval (2, 3),

find the fvalues of both the critical numbersf(0) = 4

303 202 + 3 = 3 this may be a local max/min

f(1) = 43

13 212 + 3 = 73

this may be a local max/min

2. Find the values offon the endpoints of the interval [2, 3]

End Pointsf(2) = 4

3(2)3 2(2)2 + 3 = 32

3 8 + 3 = 47

3 just plug in -2 for x, and simplify

f(3) = 43

(3)3 2(3)2 + 3 = 36 18 + 3 = 21 just plug in 3 for x, and simplify

3. The largest value from 1.&2. is the absolute maximum value,The smallest value from 1.&2. is the absolute minimum value

Make sure to add the f(x) values to the list, not the x values.Since the question asked for the abs. max. and abs. mini. values off,the answers should also be the f(x) values (as opposed to the locations).

The absolute Max is 21 (at x= 3) pick the largest number from 3, 73, 47

3 , 21

The absolute Min is 473

(at x= 2) pick the smallest number from 3, 73, 47

3 , 21

(Absolute Max/Min arent always at the endpoints)


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V =

ab

(Outer Rad.)2 (Inner Rad.)2dy

V =

2

2

(Outer Rad.)2 (Inner Rad.)2dy

V =

2

2

(y2 + 5)2 (1)2dy

V =

2

2

(y4 10y2 + 25) (1)dy

=

2

2

(y4 10y2 + 25) (1)dy

= 22

y4

10y

2

+ 24dy

= y5

5

10y3

3 + 24y

2

2

=

(2)5

5

10(2)3

3 + 24(2)

(2)5

5

10(2)3

3 + 24(2)

= 255

1023

3 + 48

25

5 + 1023

3 48

= 225

5 2102

3

3 + 2(48)

The volume of the solid created by revolving theregion bounded by the curves x = y2 + 5 andx= 1 around the y-axis is 22

5

5 2102

3

3 + 2(48).

Since I drew the typical rectangle perpendicularto the axis of revolution, I have to use the WasherMethod.This is the formula for the Washer method, itis dy because the rectangle is horizontal. Now Ineed to determine the limits of integration, and

the functions for the Inner and Outer Radii.

The limits of integration are the smallest andlargest y values of the region. (Im looking aty values, because the integral is dy .) Looking atthe picture, these clearly occur at the intersec-tion points.

Since the region is is being rotated around anaxis, the inner and outer radius are just the func-tion that bound the region at both ends of my

typical rectangle. (Since my integral is dy Iwant to make sure the function are all in theform x = an expression where the only variableisy, they already were for this question.)

Once Ive founda,b,Inner Radius and Outer Ra-dius, I need to evaluate the definite integral. Thismeans simplifying (y2 + 5)2(1)2 until I can

integrate it using the rules of Chapter 5 (specifi-cally 5.4).

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