Short26yt1 - 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

Info iconThis 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 0 ( 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 0 ( 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 r and then determine if
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background 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

This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.

Page1 / 2

Short26yt1 - 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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online