Business Calc Homework w answers_Part_15

Business Calc Homework w answers_Part_15 - Section 3.1 11...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
11. } d d y x } 5 lim h 0 } y ( x 1 h h ) 2 y ( x ) } 5 lim h 0 5 lim h 0 5 lim h 0 5 lim h 0 (4 x 1 2 h 2 13) 5 4 x 2 13 At x 5 3, } d d y x } 5 4(3) 2 13 5 2 1, so the tangent line has slope 2 1 and passes through (3, y (3)) 5 (3, 2 16). y 5 2 1( x 2 3) 2 16 y 5 2 x 2 13 12. Let f ( x ) 5 x 3 . f 9 (1) 5 lim h 0 } f (1 1 h h ) 2 f (1) } 5 lim h 0 } (1 1 h h ) 3 2 1 3 } 5 lim h 0 5 lim h 0 (3 1 3 h 1 h 2 ) 5 3 (a) The tangent line has slope 3 and passes through (1, 1). Its equation is y 5 3( x 2 1) 1 1, or y 5 3 x 2 2. (b) The normal line has slope 2 } 1 3 } and passes through (1, 1). Its equation is y 5 2 } 1 3 } ( x 2 1) 1 1, or y 5 2 } 1 3 } x 1 } 4 3 } . 13. Since the graph of y 5 x ln x 2 x is decreasing for 0 , x , 1 and increasing for x . 1, its derivative is negative for 0 , x , 1 and positive for x . 1. The only one of the given functions with this property is y 5 ln x . Note also that y 5 ln x is undefined for x , 0, which further agrees with the given graph. (ii) 14. Each of the functions y 5 sin x , y 5 x , y 5 ˇ x w has the property that y (0) 5 0 but the graph has nonzero slope (or undefined slope) at x 5 0, so none of these functions can be its own derivative. The function y 5 x 2 is not its own derivative because y (1) 5 1 but y 9 (1) 5 lim h 0 } (1 1 h h ) 2 2 1 2 } 5 lim h 0 } 2 h 1 h h 2 } 5 lim h 0 (2 1 h ) 5 2. This leaves only e x , which can plausibly be its own derivative because both the function value and the slope increase from very small positive values to very large values as we move from left to right along the graph. (iv) 15. (a) The amount of daylight is increasing at the fastest rate when the slope of the graph is largest. This occurs about one-fourth of the way through the year, sometime around April 1. The rate at this time is approximately } 2 4 4 h d o a u y rs s } or } 1 6 } hour per day. (b) Yes, the rate of change is zero when the tangent to the graph is horizontal. This occurs near the beginning of the year and halfway through the year, around January 1 and July 1. (c) Positive: January 1 through July 1 Negative: July 1 through December 31 16. The slope of the given graph is zero at x < 2 2 and at x < 1, so the derivative graph includes ( 2 2, 0) and (1, 0). The slopes at x 5 2 3 and at x 5 2 are about 5 and the slope at x 5 2 0.5 is about 2 2.5, so the derivative graph includes ( 2 3, 5), (2, 5), and ( 2 0.5, 2 2.5). Connecting the points smoothly, we obtain the graph shown. 17. (a) Using Figure 3.10a, the number of rabbits is largest after 40 days and smallest from about 130 to 200 days. Using Figure 3.10b, the derivative is 0 at these times. (b) Using Figure 3.10b, the derivative is largest after 20 days and smallest after about 63 days. Using Figure 3.10a, there were 1700 and about 1300 rabbits, respectively, at these times. 18. (a) The slope from x 5 2 4 to x 5 0 is } 0 2 2 2 ( 2 0 4) } 5 } 1 2 } . The slope from x 5 0 to x 5 1 is } 2 1 2 2 2 0 2 } 5 2 4. The slope from x 5 1 to x 5 4 is } 2 2 4 2 2 ( 2 1 2) } 5 0.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern