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MA125 CALCULUS I SPRING 2002
February 2002 TEST 1 Jellett
SCORE
NAME:..........................................
- Find the domain, and sketch the graphs, of the following functions
(i) f (x) =
x โ 2 x โ 5
(ii) f (x) =
x^2 if x โค 1
x + 1 if x > 1
- (i) For the following pair of functions, f and g, find f + g, f โ g, f g and f /g,
and state their domains.
f (x) =
2 x + 3, g(x) =
x โ 5
(ii) For the following pairs of functions, find f โฆ g, g โฆ f , f โฆ f and g โฆ g, and state
their domains.
(a) f (x) = sin x, g(x) = 2x^3 โ 1
(b) f (x) = ln x, g(x) = x
2 โ 16
(iii) Find the functions f โฆ g โฆ h and h โฆ g โฆ f when f, g, h are as follows:
f (x) =
x + 1
, g(x) = cos x, h(x) =
x + 3
- Sketch the graph of a function that satisfies all of the given conditions:
(i) lim xโ 3 +
f (x) = 4, lim xโ 3 โ
f (x) = 2, lim xโโ 2
f (x) = 2, f (3) = 3, f (โ2) = 1
(ii) f has removable discontinuities at โ 2 , 0 , 2 and jump discontinuities at โ1,
where it is left-continuous, and 1, where it is right-continuous.
- State whether or not the following limits exist and evaluate those that do.
(i) lim xโโ 3
x^2 โ x + 12
x + 3
(ii) lim xโโ 3
x
2 โ x โ 12
x + 3
(iii) lim xโ 2
x^4 โ 16
x โ 2
(iv) lim xโโ 4
|x + 4|
Bonus questions
- Let F (x) =
x^2 โ 1
|x โ 1 |
(a) Find
(i) lim xโ 1 +
F (x)
(ii) lim xโ 1 โ
F (x)
(b) Does lim xโ 1
F (x) exist?
(c) Sketch the graph of F 8. Sketch the graph of a function that satisfies the
conditions:
(i) f (0) = 0, f
โฒ (โ2) = f
โฒ (1) = f
โฒ (9) = 0, f
โฒ (0) = 1
(ii) lim xโโ
f (x) = lim xโโโ
f (x) = 0, lim xโ 6
f (x) = โโ
(iii) f โฒ(x) < 0 on (โโ, โ2), (1, 6), and (9, โ)
(iv) f โฒ(x) > 0 on (โ 2 , 1) and (6, 9)
