Prep for Calculus Notes

Math 105 - Fall 2025

Week 1 Notes

We started with a 30 minute assessment pre-test that is not for a grade. After that we reviewed the rules for working with fractions.

1. Addition. To add fractions, they need a common denominator (bottom). Once you have a common denominator, just add the numerators (tops). We did these examples:

   $$\dfrac{1}{6} + \dfrac{2}{6} \text{ and }\dfrac{1}{6} + \dfrac{2}{5}$$

2. Multiplication. I like to say that “fractions play nice with multiplication”. To multiply fractions, just multiply the tops and multiply the bottoms.

   $$\dfrac{1}{6} \cdot \dfrac{2}{5}$$

3. Division. Division is just multiplication by the reciprocal. You might have learned this as “keep-change-flip” or as “flip-and-multiply”. We did these examples:

   $$\dfrac{ ~\dfrac{1}{6}~}{~\dfrac{2}{5}~ } \text { and } \dfrac{ ~\dfrac{1}{6}~}{~5~ }.$$

Week 2 Notes

Today we reviewed order of operations and how to solve algebraic equations that don’t require factoring. You might have learned the acronym

• PEMDAS (Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction)

to remember order of operations. Another acronym is

• GEMS (Grouping, Exponents, Multiplication/Division, Addition/Subtraction).

Algebra notation can be helpful. The notation for multiplication $2 \cdot 5$ or $(2)(5)$ and division $2/5$ or $\tfrac{2}{5}$ puts numbers close together. Numbers being added or subtracted are more spread out: $2 ~ \pm ~ 5$. This is deliberate and it can help you remember that numbers being multiplied/divided are more tightly bound together and so those operations come before addition/subtraction.

1. Evaluate $\dfrac{1}{14} \left( \dfrac{1 + 2^2 \cdot 3}{7 + 2 \cdot 3} + 1 \right)^2$ (https://youtu.be/jAVbApE7lJ8)

After reviewing order of operations, we talked about:

More specifically, you can perform any algebraic operation to the expression on the left side of the equals sign as long as you also do the same operation to the right side expression. Caution: Make sure you perform the operation to everything on both sides!

2. Solve the following expression for $x$: $4 x + 2 - 6y = 0$

3. Solve $6x + 5 = 5x + 7$ (https://youtu.be/r95Yh7dMVVc)

We also talked about these two examples.

4. Simplify $\dfrac{x - 5x^2}{5x}$

5. Explain why $2(a \cdot b) \ne 2a \cdot 2b$

Week 3 Notes

We started by talking about Homework 1. When you multiply & divide fractions, look for opportunities to cancel factors whenever possible:

1. $\dfrac{24x y^2}{35} \div \dfrac{54x^3 y^2}{63 xy}$. (https://youtu.be/JIpvc0WbUBE?t=255)

We reviewed these techniques:

• Distributing $$a (b + c) = ab + ac$$

• Factoring out common factors from terms. $$ab + ac = a(b+c)$$

• Canceling common factors in fractions. $$\frac{5(x+1)}{x+1} = 5$$

• Collecting like terms. The coefficient keeps track of how many terms have been added together. $$x + 2x + 3x = 6x$$

2. Expand $2x \left(4y + \dfrac{z}{2} + x^2\right)$.

3. Factor out all common factors from $30 x^2 y + 6 x y^2 + 12 x^2 y^3$.

4. Solve $\dfrac{x}{x-1} = 3$.

5. Simplify $\dfrac{(x+3)(x+2)}{x+2}$. (https://youtu.be/lBQmy1IMko8)

Week 4 Notes

We started by reviewing some of the common mistakes that came up in the last homework. There were two that are worth mentioning.

1. If you multiply both sides of an equation by a number, make sure you distribute that number to every term!

2. When you take square roots, there are two answers, a positive one and a negative one.

We reviewed these techniques (which are opposites of each other).

• Factoring quadratic polynomials.

• FOIL-ing (First-Outside-Inside-Last).

1. Expand $(3x+2)(5x-7)$. (https://youtu.be/ZMLFfTX615w)

2. Solve $s^2 - 2s - 35 = 0$. (https://youtu.be/2ZzuZvz33X0)

We also talked about the AC-method and factoring by grouping.

3. Factor $4y^2 + 4y - 15$. (https://youtu.be/u1SAo2GiX8A)

4. How would you factor $4y^2 + 4y - 24$? Is it easier or harder than the last problem?

Week 5 Notes

Today we talked about graphing equations, and we focused on two important special cases: lines and parabolas.

1. Find an equation for the line that passes through $(-3,5)$ and $(2,1)$ in slope-intercept and point-slope form. (https://youtu.be/lzqTD0JWwhY?t=174)

2. Graph the line through $(4,-3)$ with a slope of $-2$. (https://youtu.be/5mgH-_5UJ54)

3. What are the x-values of the roots and vertex for the parabola $y = x^2 - 5x + 6$? Use those to help sketch a graph of the parabola.

4. A local elementary school wants to make a rectangular flower garden along the side of the school building. One side of the garden will the be along the wall of the school, but the other three sides will be fenced off. If you have 60 feet of fencing material, what is the largest area possible for the flower garden?

We didn’t have time for this last example, but there are a couple of problems on homework 5 that use this idea:

5. Where does the parabola $y = x^2 + 4x + 6$ intersect the line $y= -3x - 4$? (https://youtu.be/t6n-ShpFFjo)

Week 6 Notes

We started by going over questions about Homework 5 and some of the common mistakes on Homework 4. In particular we talked about the fact that a fraction is zero when its top is zero. A fraction is undefined if the bottom is zero, and we talked about why that is..

After that, we talked about function notation and graphs. We did the following examples.

1. Let $f(x) = \sqrt{25-x^2}$. Find $f(0)$, $f(3)$, $f(4)$ and $f(0)$, then graph $y = f(x)$.
   (see: https://youtu.be/ucxtbUhjwYY)

We also defined the domain and range of a function. Then we looked at different ways to express functions. For example, we had these two examples:

• The volume of a sphere: $V = \frac{4}{3} \pi r^3$.

• The function to convert Celsius to Fahrenheit: $f(x) = \tfrac{9}{5}x + 32$.

We also talked about inverse functions

2. Find the inverse of $V(r) = \tfrac{4}{3}\pi r^3$ and explain what it does in words.

3. Find the inverse of $f(x) = 2x +1$. (see: https://youtu.be/W84lObmOp8M)

Finally we looked at using the graph of a function to answer questions like the following:

4. Use the graph of $f(x) = x^4 - 3x^2 + 2$ to find all values of $x$ where $f(x) > 0$.
   (see: https://youtu.be/0-SgHGSJpKM)

Week 7 Notes

Week 8 Notes

Week 9 Notes

Week 10 Notes

Week 11 Notes

Week 12 Notes

Week 13 Notes

Week 14 Notes