Question 1.
The functions f, g and hare defined by, f(x) = ---;:2:x=+==1==, g(x) = x 2 +3 and h(x) = 2x+a.
+5x+4
a) Find the domain off.
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b) Given that (go h)(x) = 4x 2 - 8x + 7, where x =J0, calculate the value of a.
[5]
Question 2.
2.1 Find the following limits, if they exist.
a)
11. 111
v'4+/1-
h
2
.
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b)
lim
2
xIx --
4
21
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c) lim (ex + x) ¼
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.
1
d) 11111(3 - X ) 2 .
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2.2 Using the Precise definition (the c - 8 method), prove that lim (14 - 5x) = 29.
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Question 3.
a) Use the definition (first principle) to find the derivative of f(x) = Jx+T.
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b) Find the equation of the tangent line to the graph off at the point where x = 3.
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c) Find g1( x) for each of the following functions.
(i) g(x) = cos2 (cosx)
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(ii) g(x) = 3xex
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Question 4.
Consider the function f (x) = X - m if X < 3;
{ 1 - mx if x 2: 3.
a) Find the value of m for which f is a continuous function at x = 3.
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b) With the value of m you found in a), is f differentiable at x = 3 or not? Justify your answer.
[5]
1