1,001 Calculus Practice Problems

Part I

The Questions

 

Chapter 6

Exponential and Logarithmic Functions and Tangent Lines

After becoming familiar with the derivative techniques of the power, product, quotient, and chain rules, you simply need to know basic formulas for different functions. In this chapter, you see the derivative formulas for exponential and logarithmic functions. Knowing the derivative formulas for logarithmic functions also makes it possible to use logarithmic differentiation to find derivatives.

In many examples and applications, finding either the tangent line or the normal line to a function at a point is desirable. This chapter arms you with all the derivative techniques, so you'll be in a position to find tangent lines and normal lines for many functions.

The Problems You'll Work On

In this chapter, you do the following types of problems:

·        Finding derivatives of exponential and logarithmic functions with a variety of bases

·        Using logarithmic differentiation to find a derivative

·        Finding the tangent line or normal line at a point

What to Watch Out For

Although you're practicing basic formulas for exponential and logarithmic functions, you still use the product rule, quotient rule, and chain rule as before. Here are some tips for solving these problems:

·        Using logarithmic differentiation requires being familiar with the properties of logarithms, so make sure you can expand expressions containing logarithms.

·        If you see an exponent involving something other than just the variable x, you likely need to use the chain rule to find the derivative.

·        The tangent line and normal line are perpendicular to each other, so the slopes of these lines are opposite reciprocals.

Derivatives Involving Logarithmic Functions

377–385 Find the derivative of the given function.

377. f (x) = ln (x)2

378. f (x) = (ln x)4

379. 9781118496718-eq06001.eps

380. 9781118496718-eq06002.eps

381. 9781118496718-eq06003.eps

382. 9781118496718-eq06004.eps

383. 9781118496718-eq06005.eps

384. 9781118496718-eq06006.eps

385. 9781118496718-eq06007.eps

Logarithmic Differentiation to Find the Derivative

386–389 Use logarithmic differentiation to find the derivative.

386. 9781118496718-eq06008.eps

387. 9781118496718-eq06009.eps

388. 9781118496718-eq06010.eps

389. 9781118496718-eq06011.eps

Finding Derivatives of Functions Involving Exponential Functions

390–401 Find the derivative of the given function.

390. f (x) = e5x

391. 9781118496718-eq06012.eps

392. 9781118496718-eq06013.eps

393. 9781118496718-eq06014.eps

394. 9781118496718-eq06015.eps

395. 9781118496718-eq06016.eps

396. 9781118496718-eq06017.eps

397. 9781118496718-eq06018.eps

398. 9781118496718-eq06019.eps

399. 9781118496718-eq06020.eps

400. 9781118496718-eq06021.eps

401. 9781118496718-eq06022.eps

Finding Equations of Tangent Lines

402–404 Find the equation of the tangent line at the given value.

402. f (x) = 3 cos x + πx at x = 0

403. f (x) = x2 – x + 2 at (1, 2)

404. 9781118496718-eq06023.eps at x = 2

Finding Equations of Normal Lines

405–407 Find the equation of the normal line at the indicated point.

405. f (x) = 3x2 + x – 2 at (3, 28)

406. f (x) = sinx at 9781118496718-eq06024.eps

407. f (x) = 4 ln x + 2 at x = e2