Review for the MTH231 final exam

[This review will be added to along the way. If there are any answers that seem wrong, do let me know.]

It goes without saying, these questions are for practice. As they say in the movies:

This is a work of fiction. Names, characters, places and incidents either are products of the author’s imagination or are used fictitiously. Any resemblance to actual events or locales or persons, living or dead, is entirely coincidental.

By that I mean, some questions on the actual exam may resemble the ones here, some may not. In addition to going over this review, please ensure you have:

• done the WeBWorK problems – these will prepare you and improve your grade

• reviewed any quizzes

The final will be comprehensive meaning much (say 60%) will be from material you have already been tested on. The new material is:

• Section 4.7 on optimization. Be prepared for a word problem!!

• Sections 5.1 and 5.2. These define the definite integral of $f(x)$ over the interval $[a,b]$ as the area (or signed area) under the graph of $f(x)$ over $[a,b]$. There are many resulting properties of $\int_a^b f(x) dx$. Also you have an introduction to Riemann sums to approximate this area.

• Sections 5.3 and 5.4: The fundamental theorem of calculus has two parts. One part says that $\int_a^b f(x)dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$. The indefinite integral, $\int f(x) dx$, represents the collection of all antiderivatives of $f$. The other part says the derivative of an integral is the function, more precisely $(d/dx) \int_a^x f(u) du = f(x)$.

• Section 5.7 on substitution. The chain rule when reversed gives you substitution.

Practice problems on the new material

Question

Find two numbers whose sum is 26 and whose product is as large as possible.

Which formulas would you use?

$26 = x + y; p=x*y$ $26 = x*y; s = x + y$ Something else

Sum is $26$ means $26 = x+y$; product means $p=xy$

What is the value for $x$?

Is $x=y$?

Yes

No

Question

Two positive numbers multiply to $81$. Minimize the sum of the two numbers.

Which formulas would you use?

$81 = x + y; p=x*y$ $81 = x*y; s = x + y$ Something else

Product is $81$ means $81 = xy$; sum means $s = x + y$

What is the value of the smallest of the two numbers?

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We want to construct a box with a square base and we only have $10 m^2$ of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have.

What equations will you use?

$10 = 2xy +2yz + 2xz; V = xyz$ $10 = 2x^2 +4xz; V = x^2z$ $SA = 2x^2 +4xz; 10 = x^2x$ none of these choices

SA is $10$; volume to be determined

What is the value of the volume found?

$5 \sqrt{15}/9$ $\sqrt{15}/3$ $\sqrt{15}/3$

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We have a piece of cardboard that is $14$ inches by $10$ inches and we’re going to cut out the corners as shown below and fold up the sides to form a box, also shown below. Determine the height of the box that will give a maximum volume.

What formulas are useful?

$V(x) = 14 \cdot 10 \cdot x$ $V(x) = (14-x) \cdot (10-x) \cdot x$ $V(x) = (14-2x) \cdot (10-2x) \cdot x$

How many critical points are found for $V(x)$ for all $x$?

How many critical points are in the interval $[0,5]$?

What is the value of the height found?

$4 - \frac{\sqrt{39}}{3}$ $\frac{\sqrt{39}}{3} + 4$

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Find the area of the largest rectangle that fits inside a semicircle of radius $r$ (one side of the rectangle is along the diameter of the semicircle).

What equations are useful?

ERROR: invalid redefinition of constant y

What is the only positive critical point?

$x = r\sqrt{2}/2$ $x = r/2$ $x = \sqrt{r^2 - y^2}$

What is the maximum area?

$\pi r^2$ $\pi r^2/2$ $r^2$ $r^2/2$

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The figure shows $f(x) = x^2$ over the interval $[0,3]$ and:

A left-Riemann sum A midpoint-Riemann sum A right-Riemann sum

The area of the Riemann sum is?

The area of $\int_0^3 f(x) dx$ is?

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Consider this plot of some function $f$:

Is the integral $\int_1^2 f(x) dx < 0$?

Yes No

No, $f$ is positive over $[1,2]$

Is the integral $\int_0^2 f(x) dx > \int_2^4 f(x)dx$?

Yes No

Yes, the area over $[0,1]$ is greater than over $[3,4]$

Is the integral $\int_3^5 f(x) dx = 0$?

Yes No

Yes the area between $[3,4]$ cancels the area between $[4,5]$

Is $\int_1^3 f(x) dx = 2 - \pi\cdot 1^2$?

Yes No

No. Try $2 - (1/2) \pi \cdot 1^2$

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Find the following indefinite integrals:

\[ \int (x^2 + \frac{x}{10} +1)dx \]

$x^3/3 + x^2/(2\cdot 10) + x + C$ $x^3/3 + x^2/(2\cdot 10) + x$ $3x^3 + 2x^2/10 + 1x + C$ Can't do

reverse power rule

\[ \int \sin(x) dx \]

$-\cos(x) + C$ $\cos(x) + C$ $-\cos(x)$ $\cos(x)$ Can't do

need $C$

\[ \int (\sin(x) - x + \frac{x^3}{6}) dx \]

$-\cos(x) -\frac{x^2}{2} + \frac{x^4}{4\cdot 6} + C$ $-\cos(x) -\frac{x^2}{2} + \frac{x^4}{4\cdot 6}$ $-\cos(x) - 2x^2 + \frac{4x^4}{4\cdot 6} + C$ $-\cos(x) - 3x^2 + \frac{5x^4}{4\cdot 6}$ Can't do

need $C$

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Find the following definite integrals. The correct answers are numbers.

\[ \int_{-1}^1 (2x^2 + 3x + 4) dx \]

\[ \int_{0}^{\pi/2} \cos(x) dx \]

\[ \int_1^e \frac{1}{x} dx \]

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Find the "u" – specify what you would let "$u$" be to do the substitution technique:

\[ \int \sqrt{7x + 9} dx \]

$7x + 9$ $7x$ $\sqrt{x}$ None will work

\[ \int \frac{x^3}{(1 + x^4)^{1/3}}dx \]

$x^3$ $x^4$ $1 + x^4$ None will work

\[ \int x e^{x^2 + 1} dx \]

$x$ $x^2 + 1$ $x^2$ None will work

\[ \int \frac{3}{x\ln(x)} dx \]

$x$ $\ln(x)$ $x\ln(x)$ None will work

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Complete the following definite integrals using $u$-substitution.

For example:

\[ \int_{-1}^1 x^2(1-x^3)dx \]

would be approached by letting $u=1-x^3$ and then $-du/3 = x^2dx$ so:

\[ \begin{align*} \int_{-1}^1 x^2(1-x^3)dx &= -\int_{u(-1)}^{u(1)} u \frac{du}{3}\\ &= -1/3 \cdot \frac{u^2}{2} \mid_2^0 \\ &= \frac{-1}{6} (0^2 - 2^2) = \frac{4}{6}. \end{align*} \]

Now it is your turn:

\[ \int_0^{\sqrt{\pi}} x \sin(x^2) dx \]

\[ \int_0^{2} e^{5x + 2} dx \]

\[ \int_0^1 x\sqrt{1-x^2} dx \]

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