Lesson Series
The second derivative can be used to determine when a function is concave up and concave down. When $f\,''(x)\gt 0$ then the function is concave up and when $f\,''(x)\lt 0$ then the function is concave down.
An inflection point is where a function's concavity changes. We show how second derivatives can be used to find inflection points.
An integral is the area under the function $f(x)$. We can approximate the area as the sum of rectangles. There are two type of integrals: definite and indefinite.
A function's antiderivative is another function, $F(x)$, such that $F\,'(x)=f(x)$.
The Fundamental Theorem of Integral Calculus is the main technique we use to solve integrals. It states that
\[
\int_a^b\,f(x)dx=F(b)-F(a)
\]
where $F(x)$ is an antiderivative of $f(x)$.
A definite integral can be negative beause either the function is negative or because the limits of integration have been switched.
The average value of the function between $a$ and $b$ is
\[
\mbox{Average Value of }f(x)=\frac{1}{b-a}\int_a^bf(x)\,dx.
\]
We explain the exponential and natural logarithm functions.
In this lesson we state the derivatives and integrals of the exponential and natural logarithm functions. The most important result is the derivative of the natural logarithm which allows us to integrate $1/x$.
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