Welcome back to Professor Baker’s Math Class! In this session, we are shifting gears from differentiation to integration. If finding a derivative is like looking at the speedometer of a car to find its velocity at an exact moment, finding an antiderivative (or integral) is like looking at the odometer to see how far you have traveled based on that speed. We are essentially running the calculus machine in reverse.
1. The Antiderivative: Reversing the Process
Up until now, we have taken a function $f(x)$ and found its derivative $f'(x)$. Now, we are asking the question: "What function did we differentiate to get this result?"
We denote this operation with the integral symbol $\int$. The general rule involves adding a constant of integration, denoted as $+ C$, because the derivative of any constant is zero.
2. The Power Rule for Integrals
Just as we had a Power Rule for derivatives, we have one for integrals. However, instead of "multiply and subtract," we do the opposite: add and divide.
$$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $$
Important Exception: This rule works for all $n \neq -1$. If $n = -1$ (which is $\frac{1}{x}$), the power rule would result in division by zero. Instead, recall that the derivative of the natural log is $\frac{1}{x}$. Therefore:
$$ \int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln|x| + C $$
3. Initial Value Problems (Finding C)
An indefinite integral gives us a family of curves (represented by $+C$). To find the specific curve, we need an Initial Condition, such as $f(0) = -2$. By plugging in the known $x$ and $y$ values after integrating, we can solve for $C$.
Example from class:
Given $f'(x) = e^x + \frac{20}{1+x^2}$ and $f(0) = -2$, we integrate to find the general solution involving $\tan^{-1}(x)$, plug in zero, and find that $C = -3$.
4. Physics Application: Particle Motion
Calculus is the language of motion. We previously learned that differentiating position gives velocity, and differentiating velocity gives acceleration. Now, we can work backward given the acceleration:
- Acceleration $a(t)$: Integrate to find velocity.
- Velocity $v(t) = \int a(t) \, dt$: Integrate to find position.
- Position $s(t) = \int v(t) \, dt$
Note: Gravity problems often use constants like $a(t) = -32 \text{ ft/s}^2$ or $-9.8 \text{ m/s}^2$.
5. Curve Sketching and Optimization
We also revisited how derivatives help us graph functions without a calculator. By analyzing limits at infinity and the first and second derivatives, we can determine the shape of a graph.
- $f'(x) = 0$: Finds critical points (potential mins/maxs).
- $f''(x) > 0$: Concave Up (shaped like a cup).
- $f''(x) < 0$: Concave Down (shaped like a frown).
Finally, we applied these concepts to Optimization. For example, finding the maximum area of a rectangular fence given a fixed amount of material involves setting up an area equation $A(x)$, finding its derivative $A'(x)$, and setting it to zero to find the maximum dimensions.
Keep practicing those integration formulas—especially your trig functions! See you in the next class.