ELECCION DEL VEHICULO PARA EL MOVIMIENTO DE CARGA

EXAMPLE 4 Solution

dy = (3e3x _{cos x – e}3x _{sin x) dx} Multiply the derivative by dx.

= e3x _{(3 cos x – sin x) dx}

If y = ln (tan 5x), find dy.

dy = (5 sec2 _{5x dx)}

The quantity (5 sec2 _{5x dx) is the}

differential of the inside function. The definitions and some properties of dx and dy are given in this box.

DEFINITION AND PROPERTIES: Differentials Algebraically: The differentials dx and dy are defined as

dx = x dy = (x) dx

Divide both sides of the given equation by dx. Thought process:

It looks as if someone differentiated (3x + 7)6_.

But the differential of (3x + 7)6 _is

6(3x + 7)5_{· 3 · dx, or 18(3x + 7)}5 _dx.

So the function must be only 1/18 as big as (3x + 7)6_{, and the answer is}

Why is + Cneeded?

definite integral.

Q2. Write the physical meaning of derivative.

Q3. Differentiate: f(x) = 2–x

Q4. Find the antiderivative: y′ = cos x

Q5. If y = tan t where y is in meters and t is in seconds, how fast is y changing when t = /3 s?

Q6. Find lim if f(x) = sec x.

Q7. Find lim sec x.

Q8. What is the limit of a constant?

Q9. What is the derivative of a constant?

Q10. If lim g(x) = g(c), then g is —?— at x = c. A. Differentiable B. Continuous C. Undefined D. Decreasing E. Increasing

1. For f(x) = 0.2x4_{, find an equation of the linear}

function that best fits f at x = 3. What is the

error in this linear approximation of f(x) if x = 3.1? If x = 3.001? If x = 2.999?

2. For g(x) = sec x, find an equation of the linear function that best fits g at x = /3. What is the error in this linear approximation of g(x) if dx = 0.04? If dx = –0.04? If dx = 0.001?

3. Local Linearity Problem I: Figure 5-2e shows the graph of f(x) = x2 _{and the line tangent to the}

graph at x = 1.

Figure 5-2e

a. Find the equation of this tangent line, and plot it and the graph of f on the same screen. Then zoom in on the point of tangency. How does your graph illustrate that function f is locally linear at the point of tangency?

194 © 2005 Key Curriculum Press Chapter 5: Definite and Indefinite Integrals Notes: dy dx is equal to (x) EXAMPLE 5 Solution

Problem Set 5-2

Q1. Sketch a graph that illustrates the meaning of

The differential dy is usually not equal to y. y = f(x + x) – f(x)

Example 5 shows you how to find the antiderivative if the differential is given. Given dy = (3x + 7)5 _{dx, find an equation for the antiderivative, y.}

b. Make a table of values showing f(x), the

f(x) = x2 _{– 0.1(x – 1)}1/ 3

Explore the graph for x close to 1. Does the function have local linearity at x = 1? Is f differentiable at x = 1? If f is differentiable at x = c, is it locally linear at that point? Is the converse of this statement true or false? Explain.

Figure 5-2f

5. Steepness of a Hill Problem: On roads in hilly areas, you sometimes see signs like this.

Steep hill 20% grade

The grade of a hill is the slope (rise/run) written as a percentage, or, equivalently, as the number of feet the hill rises per hundred feet horizontally. Figure 5-2g shows the latter meaning of grade.

Figure 5-2g

a. Let x be the grade of a hill. Explain why the angle, degrees, that a hill makes with the horizontal is given by

b. Find an equation for d in terms of x and dx. Then find d in terms of dx for grades of x = 0%, 10%, and 20%. c. You can estimate at x = 20% by simply

multiplying d at x = 0 by 20. How much error is there in the value of found by using this method rather than by using the exact formula that involves the inverse tangent function?

d. A rule of thumb you can use to estimate the number of degrees a hill makes with the horizontal is to divide the grade by 2. Where in your work for part c did you divide by approximately 2? When you use this method to determine the number of degrees for grades of 20% and 100%, how much error is there in the number?

6. Sphere Expansion Differential Problem: The volume of a sphere is given by V = where r is the radius. Find dV in terms of dr for a spherical ball bearing with radius 6 mm. If the bearing is warmed so that its radius expands to 6.03 mm, find dV and use it to find a linear approximation of the new volume. Find the actual volume of the warmed bearing by substituting 6.03 for r. What does V equal? What error is introduced by using dV instead of V to estimate the volume?

7. Compound Interest Differential Problem: Lisa Cruz invests $6000 in an account that pays 5% annual interest, compounded continuously. a. Lisa wants to estimate how much money she

earns each day, so she finds 5% of 6000 and divides by 365. About how much will she earn the first day?

Section 5-2: Linear Approximations and Differentials © 2005 Key Curriculum Press 195 value of y on the tangent line, and the error

f(x) – y for each 0.01 unit of x from 0.97 to 1.03. How does the table illustrate that function f is locally linear at the point of tangency?

4. Local Linearity Problem II: Figure 5-2f shows the graph of

b. The actual amount of money, m, in dollars,

the first 30 days? The first 60 days?

c. Use the equation in part b to find m for the first 1 day, 30 days, and 60 days. How does the linear approximation m t dm compare to the actual value,

m = (m – 6000), as t increases?

table of sunrise times from the U.S. Naval Observatory (http://aa.usno.navy.mil/ ), the time of sunrise in Chicago for the first few days of March 2003 was changing at a rate of dS / dt –1.636 minutes per day. For March 1, the table lists sunrise at 6:26 a.m. However, no equation is given explaining how the table was computed.

a. Write the differential dS as a function of dt. Use your result to estimate the times of sunrise 10 days and 20 days later, on March 11 and March 21. How do your answers compare to the tabulated times of 6:10 a.m. and 5:53 a.m., respectively? b. Explain why dS would not give a good

approximation for the time of sunrise on September 1, 2003.

For Problems 9–26, find an equation for the differential dy. 9. y = 7x3 10. y = –4x11 11. y =(x4 _{+ 1)}7 12. y = (5 – 8x)4 13. y = 3x2 _{+ 5x – 9} 14. y = x2 _{+ x + 9} 15. y = e–1.7x 16. y = 15 ln x1/ 3 17. y = sin 3x 18. y = cos 4x 19. y = tan3 _x 20. y = sec3 _x 21. y = 4x cos x 22. y = 3x sin x 23. y = 24. y = 25. y = cos (ln x) 26. y = sin (e0.1x₎

For Problems 27–40, find an equation for the antiderivative y. 27. dy = 20x3 _dx 28. dy = 36x4 _dx 29. dy = sin 4x dx 30. dy = cos 0.2x dx 31. dy = (0.5x – 1)6 _dx 32. dy = (4x + 3)–6 _dx 33. dy = sec2 _{x dx} 34. dy = csc x cot x dx 35. dy = 5 dx 36. dy = –7 dx 37. dy = (6x2 _{+ 10x – 4)dx} 38. dy = (10x2 _{– 3x + 7)dx}

39. dy = sin5 _{x cos x dx (Be clever!)}

40. dy = sec7 _{x tan x dx (Be very clever!)}

For Problems 41 and 42, do the following. a. Find dy in terms of dx.

b. Find dy for the given values of x and dx. c. Find y for the given values of x and dx. d. Show that dy is close to y.

41. y = (3x + 4)2_{(2x – 5)}3_{, x = 1, dx = –0.04}

42. y = sin 5x, x = /3, dx = 0.06

196 © 2005 Key Curriculum Press Chapter 5: Definite and Indefinite Integrals 8. Sunrise Time Differential Problem: Based on a

in the account after the time, t, in days, is given by

m = 6000e(.05/ 365)t

Find dm in terms of t and dt. Does dm give you about the same answer as in part a for the first day, starting at t = 0 and using dt = 1? How much interest would the differential dm predict Lisa would earn in