Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice). Patrick Jones
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СКАЧАТЬ finding derivatives.

      In this chapter, you use implicit differentiation to

       Find the first derivative and second derivative of an implicit function

       Find slopes of tangent lines at given points

       Find equations of tangent lines at given points

      Lots of numbers and variables are floating around in these examples, so don’t lose your way:

       Don’t forget to multiply by dy/dx at the appropriate moment! If you aren’t getting the correct solution, look for this mistake.

       After finding the second derivative of an implicitly defined function, substitute in the first derivative in order to write the second derivative in terms of x and y.

       When you substitute the first derivative into the second derivative, be prepared to further simplify.

      408–413 Use implicit differentiation to find math.

      408. math

      409. math

      410. math

      411. math

      412. math

      413. math

      414–417 Use implicit differentiation to find math.

      414. math

      415. math

      416. math

       418–422 Find the equation of the tangent line at the indicated point.

      418. math at math

      419. math at math

      420. math at math

      421. math at math

      422. math at math

      Applications of Derivatives

      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.