Calculus exercise sheet 1 PDF

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Calculus exercise sheet...

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Exercise Sheet 1 Hand in all starred questions (both * and **) by 2pm on Thursday 09/10/14 1. Determine if the following functions are invertible and if they are find their inverses 1 1 (d) f (x) = 2 *(a) f (x) = 2x4 + 1 (b) f (x) = 2x − 1 *(c) f (x) = x +7 x+1 *2. If f (x) is an odd function and an even function, show that f (x) = 0 for all x ∈ R. 3. Sketch the functions   2 x0

**b) f (x) =



  x2 x < 0 c) f (x) = 1 x=0  x x>0

x−1 x≤ 2 3−x x>2

4. State (and commit to memory) the conditions required for the limit, lim f (x), to exist. x→a

2. *a) For f (x) defined in question 3a) find lim− f (x) and lim+ f (x). x→0

x→0

Does lim f (x) exist? If so what is it? x→0

**b) or f (x) defined in question 3b) find lim− f (x) and lim+ f (x). x→2

x→2

Does lim f (x) exist? If so what is it? x→2

a) For f (x) defined in question 3c) lim− f (x) and lim+ f (x). x→0

x→0

Does lim f (x) exist? If so what is it? x→0

5. Find the following limits or explain why they do not exist. x2 − 4 *a) lim 2 x→2 x − x − 2

x3 − 18 b) lim x→3 x − 5

d) lim 3(1 − x)(2 − x) x→2

x−5 x→2 x2 − 4

e) lim

*c) lim

h→0

**f )

√

a+h− h

√  a

2x2 + 5 . x→∞ x2 − 5x − 9 lim

*6. Given that f (x) is a function such that |f (x)| ≤ M, where M is a positive constant, for any x 6= 0, use the Sandwich Theorem to find the following limit lim xf (x). x→0...