Chapter 8
Applications of Derivatives
W
hat good are derivatives if you can’t do anything useful with them? Well, don’t worry!
There are tons and tons of useful applications involving derivatives. This chapter
illustrates how calculus can help solve a variety of practical problems, including finding
maximum and minimum values of functions, approximating roots of equations, and finding
the velocity and acceleration of an object, just to name a few. Without calculus, many of
these problems would be very difficult indeed!
The Problems You’ll Work On
This chapter has a variety of applications of derivatives, including
✓ Approximating values of a function using linearization
✓ Approximating roots of equations using Newton’s method
✓ Finding the optimal solution to a problem by finding a maximum or minimum value
✓ Determining how quantities vary in relation to each other
✓ Locating absolute and local maxima and minima
✓ Finding the instantaneous velocity and acceleration of an object
✓ Using Rolle’s theorem and the mean value theorem
What to Watch Out For
This chapter presents a variety of applications and word problems, and you may have
to be a bit creative when setting up some of the problems. Here are some tips:
✓ Think about what your variables represent in the optimization and related-rates
problems; if you can’t explain what they represent, start over!
✓ You’ll have to produce equations in the related-rates and optimization problems.
Getting started is often the most difficult part, so just dive in and try different things.
✓ Remember that linearization is just a fancy way of saying “tangent line.”
✓ Although things should be set up nicely in most of the problems, note that Newton’s
method doesn’t always work; its success depends on your starting value.
Applications of Derivatives
W
hat good are derivatives if you can’t do anything useful with them? Well, don’t worry!
There are tons and tons of useful applications involving derivatives. This chapter
illustrates how calculus can help solve a variety of practical problems, including finding
maximum and minimum values of functions, approximating roots of equations, and finding
the velocity and acceleration of an object, just to name a few. Without calculus, many of
these problems would be very difficult indeed!
The Problems You’ll Work On
This chapter has a variety of applications of derivatives, including
✓ Approximating values of a function using linearization
✓ Approximating roots of equations using Newton’s method
✓ Finding the optimal solution to a problem by finding a maximum or minimum value
✓ Determining how quantities vary in relation to each other
✓ Locating absolute and local maxima and minima
✓ Finding the instantaneous velocity and acceleration of an object
✓ Using Rolle’s theorem and the mean value theorem
What to Watch Out For
This chapter presents a variety of applications and word problems, and you may have
to be a bit creative when setting up some of the problems. Here are some tips:
✓ Think about what your variables represent in the optimization and related-rates
problems; if you can’t explain what they represent, start over!
✓ You’ll have to produce equations in the related-rates and optimization problems.
Getting started is often the most difficult part, so just dive in and try different things.
✓ Remember that linearization is just a fancy way of saying “tangent line.”
✓ Although things should be set up nicely in most of the problems, note that Newton’s
method doesn’t always work; its success depends on your starting value.