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# Week 4 Problem Set A.
Due: Wednesday 9/27
**I know it looks like there's a lot of problems.** But remember, **we learn what we do**. I just want to train you to be awesome at math, that's all. Remember, 1000 problems. I want to help you get there. Don't give up!
Write **neatly**, **organize you work**, **label section by section**. This way you can look back on it. We are not in the business of saving trees (or you can use a tablet). You can do it !
I will have office hours after class each day.
### Reading.
Please read either (or both):
- Thomas' Calculus (12E) chapters 1.1, 1.2, and 1.3
- Stewart's Calculus (9E, with Clegg and Watson) chapters 1.1, 1.2, and 1.3
Your first task is to find a copy of these (they exists somewhere). I will primarily use Thomas and supplement with Stewart. Besides the text part, these are also great sources of problems to practice from.
After doing your reading, try to put the book away and paraphrase in your own words what you have read. This helps you remember.
### Warm-up geometry.
Given the following square with a zig-zag path joining an opposing set of vertices as shown:
![[1 teaching/smc-fall-2023-math-7/week-4/---files/week-4-problems 2023-09-19 18.27.54.excalidraw.svg]]
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The path turn at right angles. Find the area of this square.
### Vertex formula for a parabola.
Given a quadratic $y = ax^{2} + bx + c$, as you know it has a graph of a parabola. It is symmetric about its vertex (the point where the parabola turns around). This vertex has $x$-coordinate $\displaystyle x= -\frac{b}{2a}$. But why?
(A) To see this, we know that the vertex is right in the middle of the two roots of the parabola. Use the **quadratic formula** and write down the expression of the $x$-coordinates of the two roots for the parabola $ax^{2}+bx+c$.
Now once you have those two $x$ values, find their average.
(B) Above method "works" but there's a slight problem / concern? Can you see what it is? Hint: We only deal with real numbers, and when we plot parabolas, everything is real.
(C) A better way is to re-write $y=ax^{2}+bx+c$ in a more suitable way by **completing the square**. Complete the square of the quadratic $ax^{2}+bx+c$. The result is sometimes called **vertex form** of the quadratic.
Hint: You can first factor out $a$, and think about what you need to add or subtract to make perfect square.
(D) Write the vertex form (hence complete the square) for the quadratic $4x^{2}-3x+11$. Then use this vertex form to help you sketch the graph of $y=4x^{2} - 3x + 11$
### Find domain and range.
In each of the following, find the domain and range of each. For some, you might want to think about the graph of the function, of parts of the function (as done in class), to determine the range.
1. $f(x)=\sqrt{5x+10}$
2. $\displaystyle g(t)=\frac{4}{3-t}$
3. $h(x)= \sqrt{x^{2}-5x}$
4. $\displaystyle j(x)= \frac{2}{x^{2}-16}$
5. $\displaystyle k(x) = 2 + \frac{x^{2}}{x^{2}+4}$
6. Just find the domain of $\displaystyle y= \frac{x+3}{4-\sqrt{x^{2}-9}}$
### Finding formulas for functions.
1. Express the side length $L$ of a square as a function of the length $d$ of the square's diagonal. Then express the area $A$ as a function of the diagonal $d$.
2. A point $P$ in the first quadrant lies on the graph of the function $f(x)=\sqrt{x}$. Express the $x$- and $y$-coordinates of $P$ as functions of the slope $m$ of the line joining $P$ to the origin.
- Hint: Draw a diagram first, label the point $P$ on the graph of $f(x)=\sqrt{x}$. Express the slope $m$ of the line from $P$ to the origin as a function of $x$ first, then solve $x$ in terms of $m$.
3. Consider the point $(x,y)$ lying on the graph of $y=\sqrt{x-3}$. Let $L$ be the distance between the points $(x,y)$ and $(4,0)$. Write $L$ as a function of $y$.
### Functions and graphs.
Find the domain, range, and graph the following:
1. $f(x)=\sqrt{x}$
2. $f(x) = \sqrt{|x|}$
3. $f(x)=\sqrt{-x}$
4. $\displaystyle g(x)=\frac{1}{x}$
5. $\displaystyle g(x)=\frac{1}{|x|}$
6. $\displaystyle g(x)= \frac{x}{|x|}$
### Piecewise-defined functions.
Graph, and find the domain and range for each of the following. Pay attention where the function is defined. A question to keep in mind is "do these pieces join up?":
1. $\displaystyle f(x) = \begin{cases} x, & 0\le x\le1 \\ 2-x, & 1 < x \le 2\end{cases}$
2. $\displaystyle g(x) = \begin{cases} 1 - x, & 0\le x\le1 \\ 2-x, & 1 < x \le 2\end{cases}$
3. $\displaystyle h(x) = \begin{cases} 4-x^{2}, & x \le 1 \\ x^{2}+2x, & x > 1\end{cases}$
4. $\displaystyle k(x) = \begin{cases} 1/x, & x < 0 \\ x, & 0 \le x\end{cases}$
### Increasing and decreasing functions.
For each of the following functions, give a sketch of its graph, and indicate the intervals over which the function is increasing and the intervals where it is decreasing.
1. $y = -x^{3}$
2. $\displaystyle y= -\frac{1}{x^{2}}$
3. $\displaystyle y= - \frac{1}{x}$
4. $\displaystyle y = \frac{1}{|x|}$
5. $y = \sqrt{|x|}$
6. $y= - 3 \sqrt{x}$
7. $\displaystyle y = -\frac{3}{\sqrt{x}}$
### Even and odd functions.
For each of the following, say whether the function is even, odd, or neither. (Review what it means algebraically and graphically.) Give reasons for your answer.
1. $f(x) = 3$
2. $f(x)=x^{2} + 1$
3. $f(x)=(x+1)^{2}$
4. $g(x)=x^{3}+x$
5. $g(x) = x^{3}+x^{2}$
6. $\displaystyle h(x)= \frac{1}{x^{2}-1}$
7. $\displaystyle h(x) = \frac{1}{x-1}$
8. $k(x) = x^{3}$
9. $k(x) = |x^{3}|$
10. $j(x) = 2x + 1$
11. $j(x) = 2|x| +1$
### Theory and practice.
Reading comprehension is an important part of math. Word problems are real problems.
1. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions $14$ in. by $22$ in. by cutting out equal squares of side $x$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $x$ ![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230920034739.png]]
2. The accompanying figure shows a rectangle inscribed in an isoceles right triangle whose hypotenuse is $2$ units long.
- (A) Express the $y$-coordinate of $P$ in terms of $x$. (Hint: Start by writing an equation for the line $AB$)
- (B) Express the area $A$ of the rectangle in terms of $x$.![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230920034946.png]]
3. Algebraically find all values of $x$ for which $$
\frac{x}{2} > 1 + \frac{4}{x}
$$ is true. (Hint: Pretend it is an equality first, identify all the points of interest on a number line, and then check each interval whether the inequality is satisfied or not).
- After you have done this, use a graphing technology like DESMOS to plot $y=\frac{x}{2}$ and $y = 1+ \frac{4}{x}$ to confirm your answer.
4. Algebraically find all values of $x$ for which $$
\frac{3}{x-1} < \frac{2}{x + 1}
$$ is true. (Hint: Pretend it is an equality first, identify all the points of interest on a number line, and then check each interval whether the inequality is satisfied or not).
- After you have done this, use a graphing technology like DESMOS to plot $y=\frac{3}{x-1}$ and $y = \frac{2}{x + 1}$ to confirm your answer.
5. **Industrial cost.** A power plant typically wants to be next to a river for cooling purposes. Here a power plant sits next to a river that is $800$ ft wide. To lay a new cable from the plant to a location in the city $2$ mi. downstream on the opposite side costs $\$180$ per foot across the river and $\$100$ per foot along the land. (Note 1 mi. = 5280 ft.)![[1 teaching/smc-fall-2023-math-7/week-4/---files/Pasted image 20230920040703.png]] Suppose that the cable goes from the plant to a point $Q$ on the opposite side that is $x$ ft. from the point $P$ directly opposite the plant. Write a function $C(x)$ that gives the cost of laying the cable in terms of the distance $x$.
//// You did it! Great job! ////