Fall 2025: Chapters 4-9 and 5-1 Class Notes

Welcome to a review of key concepts from Chapters 4-9 and 5-1! Let's explore the fundamental ideas of derivatives and integrals with some helpful examples and explanations.

Derivatives: A Quick Recap

Remember, the derivative of a function, denoted as $f'(x)$, gives us the instantaneous rate of change of the function. Here are some examples:

  • If $f(x) = \frac{1}{3}x^3$, then $f'(x) = x^2$.
  • If $f(x) = -\cos(x)$, then $f'(x) = \sin(x)$.
  • If $f(x) = \ln|x|$, then $f'(x) = \frac{1}{x}$.

Antiderivatives: Reversing the Process

The antiderivative, also known as the indefinite integral, is the reverse operation of differentiation. We seek a function $F(x)$ such that $F'(x) = f(x)$. Note that antiderivatives include a constant of integration, $C$, because the derivative of a constant is always zero.

For example, if $f'(x) = x^2$, then $f(x) = \frac{1}{3}x^3 + C$. The $+C$ accounts for the fact that the derivative of $\frac{1}{3}x^3 + 5$ is also $x^2$, as is the derivative of $\frac{1}{3}x^3 - \pi$. Therefore, we need to add $C$ to account for any constant.

Important Antiderivative Formulas

Here's a table summarizing some essential antiderivative formulas. Remember these, as they will be incredibly useful! These are also called Indefinite Integrals.

  • The antiderivative of $\sin(x)$ is $-\cos(x) + C$.
  • The antiderivative of $\cos(x)$ is $\sin(x) + C$.
  • The antiderivative of $\frac{1}{x}$ is $\ln|x| + C$.

The Integral Power Rule

The integral power rule is a fundamental tool for finding antiderivatives of power functions. It states:

If $f'(x) = x^n$, then $f(x) = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.

Let's see it in action: To find the antiderivative of $x^2$, we apply the power rule:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

So, $\int x^2 dx = \frac{x^{2+1}}{2+1} + C = \frac{1}{3}x^3 + C$

Definite Integrals

While indefinite integrals give us a family of functions (due to the constant $C$), definite integrals give us a specific value, representing the area under a curve between two points. A definite integral is expressed as:

$\int_a^b f(x) dx$

For example, let's evaluate $\int_0^1 x^2 dx$:

First, find the antiderivative: $\frac{1}{3}x^3$.

Then, evaluate the antiderivative at the upper and lower limits of integration: $\frac{1}{3}(1)^3 - \frac{1}{3}(0)^3 = \frac{1}{3} - 0 = \frac{1}{3}$.

Therefore, $\int_0^1 x^2 dx = \frac{1}{3}$.

Keep practicing, and you'll master these concepts in no time! Remember, calculus is all about understanding change and accumulation. Good luck, and keep exploring!