Patrick Jones - Calculus - 1001 Practice Problems For Dummies (+ Free Online Practice)

Chapter 8

What 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!

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

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.

423–425Find the differential dy and then evaluate dy for the given values of x and dx.

423. ,

424. , ,

425. , ,

426–428Find the linearization L(x) of the function at the given value of a.

426.

427. ,

428. ,

429– 431Estimate the value of the given number using a linearization.

429. Estimate 7.96 2/3to the thousandths place.

430. Estimate to the tenths place.

431. Estimate tan 46° to the thousandths place.

432– 445Solve the related-rates problem. Give an exact answer unless otherwise stated.

432. If V is the volume of a sphere of radius r and the sphere expands as time passes, find in terms of .

433. A pebble is thrown into a pond, and the ripples spread in a circular pattern. If the radius of the circle increases at a constant rate of 1 meter per second, how fast is the area of the circle increasing when the radius is 4 meters?

434. If and , find when .

435. If , , and , find when and .

436. Two sides of a triangle are 6 meters and 8 meters in length, and the angle between them is increasing at a rate of 0.12 radians per second. Find the rate at which the area of the triangle is increasing when the angle between the sides is . Round your answer to the nearest hundredth.

437. A ladder 8 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 3 feet per second, how fast is the angle between the top of the ladder and the wall changing when the angle is radians?