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, ex.

·        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. 9781118496718-eq12001.eps

751. 9781118496718-eq12002.eps

752. 9781118496718-eq12003.eps

753. 9781118496718-eq12004.eps

754. 9781118496718-eq12005.eps

755. 9781118496718-eq12006.eps

Note: The derivative formula for sec−1 t varies, depending on the definition used. For this problem, use the formula .png

756. y = csc−1 e2x

757. 9781118496718-eq12008.eps

Note: The derivative formula for sec−1 x varies, depending on the definition used. For this problem, use the formula 9781118496718-eq12009.eps.

758. 9781118496718-eq12010.eps

759. 9781118496718-eq12011.eps

760. 9781118496718-eq12012.eps

761. 9781118496718-eq12013.eps

762. 9781118496718-eq12014.eps

Finding Antiderivatives by Using Inverse Trigonometric Functions

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

763. 9781118496718-eq12015.eps

764. 9781118496718-eq12016.eps

765. 9781118496718-eq12017.eps

766. 9781118496718-eq12018.eps

767. 9781118496718-eq12019.eps

768. 9781118496718-eq12020.eps

769. 9781118496718-eq12021.eps

770. 9781118496718-eq12022.eps

771. 9781118496718-eq12023.eps

772. 9781118496718-eq12024.eps

773. 9781118496718-eq12025.eps

774. 9781118496718-eq12026.eps

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. 9781118496718-eq12027.eps

Finding Derivatives of Hyperbolic Functions

780–789 Find the derivative of the given function.

780. y = coshx

781. 9781118496718-eq12028.eps

782. 9781118496718-eq12029.eps

783. 9781118496718-eq12030.eps

784. y = tanh(sinh x)

785. 9781118496718-eq12031.eps

786. 9781118496718-eq12032.eps

787. 9781118496718-eq12033.eps

788. 9781118496718-eq12034.eps

789. 9781118496718-eq12035.eps

Finding Antiderivatives of Hyperbolic Functions

790–799 Find the antiderivative.

790. 9781118496718-eq12036.eps

791. 9781118496718-eq12037.eps

792. 9781118496718-eq12038.eps

793. 9781118496718-eq12039.eps

794. 9781118496718-eq12040.eps

795. 9781118496718-eq12041.eps

796. 9781118496718-eq12042.eps

797. 9781118496718-eq12043.eps

798. 9781118496718-eq12044.eps

799. 9781118496718-eq12045.eps

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. 9781118496718-eq12046.eps

801. 9781118496718-eq12047.eps

802. 9781118496718-eq12048.eps

803. 9781118496718-eq12049.eps

804. 9781118496718-eq12050.eps

805. 9781118496718-eq12051.eps

806. 9781118496718-eq12052.eps

807. 9781118496718-eq12053.eps

808. 9781118496718-eq12054.eps

809. 9781118496718-eq12055.eps

810. 9781118496718-eq12056.eps

811. 9781118496718-eq12057.eps

812. 9781118496718-eq12058.eps

813. 9781118496718-eq12059.eps

814. 9781118496718-eq12060.eps

815. 9781118496718-eq12061.eps

816. 9781118496718-eq12062.eps

817. 9781118496718-eq12063.eps

818. 9781118496718-eq12064.eps

819. 9781118496718-eq12065.eps

820. 9781118496718-eq12066.eps

821. 9781118496718-eq12067.eps

822. 9781118496718-eq12068.eps

823. 9781118496718-eq12069.eps

824. 9781118496718-eq12070.eps

825. 9781118496718-eq12071.eps

826. 9781118496718-eq12072.eps

827. 9781118496718-eq12073.eps

828. 9781118496718-eq12074.eps

829. 9781118496718-eq12075.eps

830. 9781118496718-eq12076.eps

831. 9781118496718-eq12077.eps