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problem 1

(a)  Find the natural domain and determine the range of the following functions.

i. f (x) = √x2 - 3

ii. f (x) = 3sin x

(b)   Sketch the graph of the function:

 y = 1 + 2x - x2

by appropriately translating and reflecting the function y = x2

(c)  Find expressions for

f (x) = x/1 + x2, g (x) = 1/x

Find the domains of both the compositions.
(d)  Using the horizontal line test find which of the following functions f is one-to-one:

(i)  f (x) = 3x + 2

(ii)  f (x) = x


(a)  Find the value of:


(b)  Determine the value of:


(c)  (i)  Find a value of the constant k, which will make the following function continuous.


(ii)  Sketch the continuous function.
(iii)  Show that the continuous function can be written as a polynomial.

(a)  Using the definition of the derivative,  dx/df of a function f(x) with respect to x as:


(b)  Find dx/dy of: y = (3x2 + 2/x + 3)

(c)  A car is driven down a street highway such that in the first 12 seconds it travels a distance s = 4.5t2 m from its initial position.

(i)  Find the average velocity of the car over the time interval [0,12] s.

(ii)  Find its instantaneous velocity at t = 6 s.

(d)  A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15  cm/min. At what rate (cm3/min) must air be removed when the radius is 9 cm?

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