## 1,001 Calculus Practice Problems

__Part I__

## The Questions

__Chapter 12__

### Inverse Trigonometric Functions, Hyperbolic Functions, and L'Hôpital's Rule

This chapter looks at the very important inverse trigonometric functions and the hyperbolic functions. For these functions, you see lots of examples related to finding derivatives and integration as well. Although you don't spend much time on the hyperbolic functions in most calculus courses, the inverse trigonometric functions come up again and again; the inverse tangent function is especially important when you tackle the partial fraction problems of Chapter __14__. At the end of this chapter, you experience a blast from the past: limit problems!

*The Problems You'll Work On*

This chapter has a variety of limit, derivative, and integration problems. Here's what you work on:

· Finding derivatives and antiderivatives using inverse trigonometric functions

· Finding derivatives and antiderivatives using hyperbolic functions

· Using L'Hôpital's rule to evaluate limits

*What to Watch Out For*

Here are a few things to consider for the problems in this chapter:

· The derivative questions just involve new formulas; the power, product, quotient, and chain rules still apply.

· Know the definitions of the hyperbolic functions so that if you forget any formulas, you can easily derive them. They're simply defined in terms of the exponential function, *e ^{x}*.

· Although L'Hôpital's rule is great for many limit problems, make sure you have an indeterminate form before you use it, or you can get some very incorrect solutions.

*Finding Derivatives Involving Inverse Trigonometric Functions*

*750–762** Find the derivative of the given function.*

**750.**

**751.**

**752.**

**753.**

**754.**

**755.**

** Note:** The derivative formula for sec

^{−1}

*t*varies, depending on the definition used. For this problem, use the formula

**756.** *y* = csc^{−1}* e*^{2x}

**757.**

** Note:** The derivative formula for sec

^{−1}

*x*varies, depending on the definition used. For this problem, use the formula .

**758.**

**759.**

**760.**

**761.**

**762.**

*Finding Antiderivatives by Using Inverse Trigonometric Functions*

*763–774** Find the indefinite integral or evaluate the definite integral.*

**763.**

**764.**

**765.**

**766.**

**767.**

**768.**

**769.**

**770.**

**771.**

**772.**

**773.**

**774.**

*Evaluating Hyperbolic Functions Using Their Definitions*

*775–779** Use the definition of the hyperbolic functions to find the values.*

**775.** sinh 0

**776.** cosh (ln 2)

**777.** coth (ln 6)

**778.** tanh 1

**779.**

*Finding Derivatives of Hyperbolic Functions*

*780–789** Find the derivative of the given function.*

**780.** *y* = cosh^{2 }*x*

**781.**

**782.**

**783.**

**784.** *y* = tanh(sinh *x*)

**785.**

**786.**

**787.**

**788.**

**789.**

*Finding Antiderivatives of Hyperbolic Functions*

*790–799** Find the antiderivative.*

**790.**

**791.**

**792.**

**793.**

**794.**

**795.**

**796.**

**797.**

**798.**

**799.**

*Evaluating Indeterminate Forms Using L'Hôpital's Rule*

*800–831** If the limit is an indeterminate form, evaluate the limit using L'Hôpital's rule. Otherwise, find the limit using any other method.*

**800.**

**801.**

**802.**

**803.**

**804.**

**805.**

**806.**

**807.**

**808.**

**809.**

**810.**

**811.**

**812.**

**813.**

**814.**

**815.**

**816.**

**817.**

**818.**

**819.**

**820.**

**821.**

**822.**

**823.**

**824.**

**825.**

**826.**

**827.**

**828.**

**829.**

**830.**

**831.**