Math 140 - Week 4 Notes

Monday, March 8

Today we focused on continuity. We talked about the three main types of discontinuities that graphs might have:

Hole Discontinuities

1. Simplify the function $$\displaystyle \frac{x^2 + 2x + 1}{x+1}$$ and then graph it. Notice that there is a hole discontinuity in the graph at the bad point when $$x = -1$$.

Jump Discontinuities

1. Graph the function $$y = \dfrac{x}{|x|}$$. Notice that there is a jump discontinuity at $$x = 0$$ where the y-value suddenly jumps from being $$-1$$ to $$+1$$.

Here is a real world example of a jump discontinuity:

1. A company will reimburse driving mileage at a rate of 50 cents per mile for trips under 100 miles, but only 30 cents per mile for trips over 100 miles. Graph the reimbursement amount $$R$$ as a function of travel distance $$x$$.

Pole Discontinuities

1. The function $$f(x) = \dfrac{6}{x-3}$$ has a vertical asymptote at $$x = 3$$. This is called a pole discontinuity.

Limits

The limit of a function $$y = f(x)$$ as $$x$$ approaches $$a$$ is the $$y$$-value that the graph appears to be heading towards. We use the notation $$\displaystyle \lim_{x \rightarrow a} f(x)$$ to represent the limit. To calculate a limit, use the following steps:

1. First try plugging in $$a$$ into the function. If $$f$$ is continuous at $$a$$, then the $$y$$-value you get is the answer.

2. If $$f(a)$$ is undefined, try to simplify the function using algebra until you can calculate a value at $$a$$. As soon as you get a number, you are done.

3. If $$f(x)$$ is heading towards more than one y-value, then we say that the limit does not exist (DNE for short).

We did the following examples:

1. $$\displaystyle \lim_{x \rightarrow 2} \frac{x^2 - 3x +2}{x-2}$$

2. $$\displaystyle \lim_{x \rightarrow 3} \frac{x^2 - 3x +2}{x-2}$$ (This time, $$x=3$$ isn’t a bad point…)

3. $$\displaystyle \lim_{x \rightarrow 0} \frac{x-4}{x}$$ (Notice that when you plug-in $$x=0$$, you get a non-zero divided by zero…)

We finished by discussing one-sided limits. We did these two examples:

1. $$\displaystyle \lim_{x \rightarrow 0^-} \frac{x-4}{x}$$ = \$.

Think of $$0^-$$ as a very tiny, but negative number. So the limit above is $$+\infty$$. Try finding:

1. $$\displaystyle \lim_{x \rightarrow 0^+} \frac{x-4}{x}$$.

Wednesday, March 10

We started by reviewing limits. We did the following problems in class:

1. $$\displaystyle \lim_{x \rightarrow 2} \frac{x^2 - 4}{x-2}$$ (Answer is 4)

2. $$\displaystyle \lim_{x \rightarrow 3^-} \frac{1}{(x-3)^2}$$ (Answer is $$\infty$$)

3. $$\displaystyle \lim_{x \rightarrow 1} \frac{x^2-1}{x^2 + 1}$$ (Answer is 0)

Derivatives

Then we introduced the derivative of a function $$y = f(x)$$. The derivative of $$f$$ at $$x$$ is two things:

• It’s the slope of a tangent line at $$x$$.
• It’s the instantaneous rate of change.

We use several different notations to represent the derivative: $$f'(x)$$, and $$y'$$, and $$\dfrac{dy}{dx}$$. These all mean the same thing.

To calculate the derivative, today we used the definition of the derivative: $f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}.$ We did these two examples in class:

1. Use the definition of the derivative to find the derivative of $$f(x) = x^2$$.

2. Use the definition of the derivative to find the derivative of $$y = x^3$$.

Friday, March 12

Today we talked about easier ways to calculate derivatives. Here are three important rules. The symbol $$\dfrac{d}{dx}$$ is shorthand that literally means “take the derivative with respect to $$x$$.”

1. Power Rule. $$\dfrac{d}{dx} x^p = p x^{p-1}$$.

2. Sum Rule. $$\dfrac{d}{dx} f(x) + g(x) = f'(x) + g'(x)$$.

3. Constant Multiple Rule. $$\dfrac{d}{dx} c f(x) = c f'(x)$$.

We used the power rule to differentiate $$x^5, \sqrt{x}$$, and $$\dfrac{1}{x}$$. Then we calculated the following examples in class:

1. $$\dfrac{d}{dx} x^2 - 4x + 4$$.

Notice that the derivative of $$-4x$$ is just $$-4$$ since that is the slope of the line $$y= -4x$$, and the derivative of any constant term like $$4$$ is just zero.

1. $$\dfrac{d}{dx} x \sqrt{x}$$.

Hint: Use algebra to find the combined power of $$x$$ first.

1. $$\dfrac{d}{dx} \dfrac{1}{x} - \dfrac{8}{x^2}$$.

Then we talked about the meaning of derivatives.

1. If a rock is dropped, it falls $$s$$ feet in $$t$$ seconds where $$s(t) = 16t^2$$. Find the derivative $$s'(t)$$. What are the units of the derivative and what does it represent?

2. Suppose it costs $$C(x) = 100 + \sqrt{x}$$ dollars to make $$x$$ golf balls. The marginal cost is the cost of making one more golf ball after producing $$x$$ of them. It is approximately the same as the derivative $$C'(x)$$. We calculated $$C'(x)$$ and then used it to verify that $$C'(25)$$ really is about the same as the cost of making one more golf ball after already having made 25.

3. Suppose that a town’s population will grow according to the formula $$P(t) = 4000 + 10t^3$$, where $$t$$ is measured in years from now. Find $$P'(t)$$. What are the units of $$P'(t)$$ and what does $$P'(t)$$ represent?