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short26yt1

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

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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

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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....
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