Derivative applications

Equations of normal and tangent lines

A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, we'll not only find equations of tangent lines, but normal ones as well.
Equation of tangent line example 1
Use the derivative to find the equation for a tangent line.
y-intercept of tangent line example
A practice problem for finding the y-intercept of a tangent line to f(x) = 1/x
Equation of normal line
The "normal" line of a curve at a point is perpendicular to the tangent line.  Here we practice finding the equation for a normal line.
Applications of derivatives: Tangent and normal lines
Practice finding tangent and normal lines. 

Critical points and graphing with calculus

One of the reasons calculus was invented was to be able to optimize functions. When you have some function modeling a real world situation, you often want to find its maximum or minimum. In this tutorial, you will see how information about the derivative of a function can give powerful ways to mathematically describe the "shape" of a function.
Minima, maxima and critical points
Here we see how the highest and lowest points on a graph relate to the derivative. 
Finding critical numbers
Critical numbers of a function are inputs where the derivative equals zero.  Here we practice finding some.
Critical numbers
Practice finding critical numbers.
Testing critical points for local extrema
Once you find a critical point, how can you tell if it is a minimum, maximum or neither?
Identifying minima and maxima for x^3 - 12x + 2
Some practice with finding extrema of a function.

Absolute and relative maxima and minima

When you're looking for the maximum or minimum of a function, a good way to start is by finding points where the derivative equals zero. However, you won't always get the maximum possible value of the function; you might just end up with a point which is maximum *relative* to those points around it. In this tutorial, you will learn about the extreme value theorem, and what it tells us about relative maxima and minima.
Extreme value theorem
The extreme value theorem tells us that continuous functions must attain a maximum and a minimum on any closed interval where it is defined.
Introduction to minimum and maximum points
Sal explains all about minimum and maximum points, both absolute and relative.
Identifying relative minimum and maximum values
How can you tell if a point is a miximum/minimum of a function?
Extreme values from graphs
Practice identifying maximum and minimum values by looking at the graph of a function.
Applying extreme value theorem
Watch the extreme value theorem in action.
Extreme value theorem
Practice using the extreme value theorem.

Tangents to polar curves

Here you have the chance to practice thinking about tangent lines when curves are defined in polar coordinates.
Tangents to polar curves
Practice tangent lines of functions defined with polar coordinates.