Antiderivatives Analytical
From last time (7.0)...
- Handout for 7.0 (.pdf)
- Antiderivatives [7.0] (.ppt)
Antiderivative rules
| Rule | Derivative | $\Rightarrow$ Antiderivative |
|---|---|---|
| Constant | $\left(c\right)'=0$ | $\int 0\,dx=0 + C$ |
| Power | $\left(x^n\right)'=nx^{n-1}$ | $\int x^n\,dx=\frac{1}{n+1}x^{n+1} + C$ |
| Exponential | $\left(e^x\right)'=e^x$ | $\int e^x\,dx=e^x + C$ |
| Logarithm | $\left( \ln x \right)'=\frac1x$ | $\int \frac 1x\,dx=\ln x + C$ |
| Sine | $\left( \sin x \right)'= \cos x$ | $\int \cos x\,dx= \sin x + C$ |
| Constant multiple | $\left( cf(x)\right)'=c f'(x)$ | $\int c f(x)\,dx=c\int f(x)\,dx$ |
| Sum and Difference | $\left( f(x)\pm g(x) \right)'=f'(x)\pm g'(x)$ | $\int \left(f(x)+g(x)\right) \,dx=\int f(x)\,dx+\int g(x)\,dx$ |
| Chain rule (substitution) | $\left( f(g(x))\right)'=f'(g(x))\,g'(x)$ | $\int f'(g(x))\,g'(x)\,dx = \int f'(g)\,dg=f(g(x))+C$ |
| Sum and Difference | $\left( f(x)\pm g(x) \right)'=f'(x)\pm g'(x)$ | $\int \left(f(x)\pm g(x)\right) \,dx=\int f(x)\,dx\pm\int g(x)\,dx$ |
| Product rule | $\left( f(x)g(x)\right)'=f'(x)g(x) +f(x)g'(x)$ | ${\color{#888}\int \left(f(x)g(x)\right)'\,dx=\ \ }f(x)g(x)=$ $\ \ \ \ \ \int f'(x)g(x)\,dx+\int f(x)g'(x)\,dx$ |
Homework hints
Problem 9
To find the area between the graph of $f(x)$ and the $x$ axis: Graph the function, and you'll find the graph goes below the $x$ axis. Express the area in terms of a definite integral, to capture just the area below the x-axis.
Finally, you know that if $f(x)$ is below the $x$ axis, then the definite integral will be negative. But the Edfinity problem just asks for the area, by which they mean the (positive) magnitude of the area.
Problem 10
You need to calculate the average value from 1 to $c$ of a linear function (maybe $f(x)=3x$ ). This is... $$ \left(\text{average } f \right)_{1\lt x\lt c}=\frac{1}{c-1}\int_1^c f(x)\,dx$$ You know that the definite integral in this expression represents the area under the graph of $f(x)$ for $1\lt x \lt c$. This $f(x)$ is a linear (straight line) function. Make an accurate graph of the function for $1\lt x \lt c$, and I think you'll see how you can calculate the "area under the graph" exactly using rectangles and triangles.