Linear approximation module
After completing this section, students should be able to do the following.
- Define linear approximation as an application of the tangent to a curve.
- Find the linear approximation to a function at a point and use it to approximate the function value.
- Identify when a linear approximation can be used.
- Label a graph with the appropriate quantities used in linear approximation.
- Find the error of a linear approximation.
- Compute differentials.
- Use the second derivative to discuss whether the linear approximation over or underestimates the actual function value.
- Contrast the notation and meaning of dydy versus ΔyΔy.
- Understand that the error shrinks faster than the displacement in the input.
- Justify the chain rule via the composition of linear approximations.
Replacing curves with lines
Two young mathematicians discuss linear approximation.
We use a method called ``linear approximation'' to estimate the value of a (complicated) function at a given point.
Explanation of the product and chain rules
We give explanation for the product rule and chain rule.