1E1 Christmas exam 2004
1. Calculate the following limits, provided they exist. Otherwise explain why
they fail to exist.
¡
¢
(a) lim x3 + 4x2 − 3x − 2
x→−2
√
√
2x − x + 1
(b) lim
x→1
x−1
¶
µ
1
3
cos x
(c) lim
x→∞ x2
x2 − 4
(d) lim
x→−2 x + 2
x
(e) lim
x→0 x + 2 |x|
2. Calculate the following derivatives.
(a)
(b)
(c)
(d)
(e)
d p
4
x3 − x + 1
dx
d
(sin(x) cos(x))
dx µ
¶
d 3x2 − 2x + 1
dx
x−4
d
cot(2x3 )
dx
¢
d ¡√
x cos(3x)
dx
(x 6= 4)
3. Let f be a function defined on (−∞, ∞) by
f (x) = x4 − 8x2 .
(a) Show that
f 0 (x) = x(4x2 − 16).
(b) Show that
f 00 (x) = 12x2 − 16.
(c) Find all minima and maxima of f .
√
(d) Is there a point of inflection at x = 2/ 3? Give reasons for your
answer.
√
(e) Is there a point of inflection at x = −2/ 3? Give reasons for your
answer.
4. Use implicit differentiation to answer the following questions.
1
(a) Consider the curve given by
x3 + y 3 − 9xy = 0.
Determine the slope-intercept equations of the tangent at (2, 4), and
the line passing through (2, 4) which is perpendicular to the tangent.
(b) Let f be a function that is defined on an open interval D. Let f be
differentiable with f 0 (x) 6= 0 for all x in D. Then f is one-to-one and
thus invertible. Let R be the range of f . Show that
(f −1 )0 (y) =
with y = f (x), for all y in R.
2
1
f 0 (x)