{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

short26yt1 - r< 1 or r> 1[8 7 Let f x = x 3 x 1(a Show...

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

View Full Document Right Arrow Icon
Physical Sciences Division University of Toronto at Scarborough MATA26Y October 28, 1996 110 minutes TERM TEST I [20] 1. (a) Compute the derivative f ( x ) for each of the following functions f ( x ). Note: Simplification of your answer is not required. (i) f ( x ) = 5 x + 3 2 x 2 + 7 (ii) f ( x ) = x 1 / 3 (3 x 2 + 1) (iii) f ( x ) = x 2 + x + 3 (iv) f ( x ) = ( x 2 + 1 + 2 x ) 25 [5] (b) Use the definition of derivative to compute the derivative f ( x ) for the func- tion f ( x ) = 1 /x . Note: In part (b), NO credit will be given for computations which use the power rule rather than the definition of derivative. [4] 2. Find the interval(s) on which f ( x ) = x 3 3 - 5 x 2 2 + 4 x + 1 is increasing. [5] 3. (a) Find all values of x which satisfy 3 ( x 2 ) = 9 x . [8] (b) Suppose a bacterial culture grows exponentially. Assume that initially it contains 2000 organisms and that the amount doubles every 40 minutes. How long will it take until there are 10000 organisms? [10] 4. Find the equation of the tangent line to f ( x ) = x 3 + 2 x + 10 at the point where x = 2. [5] 5. Find sin(tan - 1 ( x + 1)). [10] 6. Show that x 5 + x - 1 has precisely one root
Image of page 1

Info icon This 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.

Unformatted text preview: r < 1 or r > 1. [8] 7. Let f ( x ) = x 3 + x + 1. (a) Show that f ( x ) is invertible. (b) Compute g (3) where g is the inverse function to f . [8] 8. Find all values of x which satisfy the following inequalities. (a) 2 x x-1 > 1 (b) 2 x | x-1 | > 1 MATA26Y page 2 [8] 9. Let f ( x ) = | x-1 | -1. State the hypotheses and conclusion of Rolle’s theorem and explain why this f ( x ) does not contradict the theorem even though f (0) = 0, f (2) = 0, and there is no z in (0 , 2) for which f ( z ) = 0. (You may accept the preceding statement about the derivative of f as fact; you do not have to prove it.) [10] 10. Let f ( x ) = 1 x-1 . (a) Find the linear approximation L ( x ) based at x = 3 to f ( x ) . (b) Find an interval (3-h, 3+ h ) containing 3 throughout which the error in ap-proximating f ( x )by its linear approximation based at x = 3 is less than 0 . 001....
View Full 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