MA1S11 Tutorial Sheet 91
8-11 December 2015
Useful facts:
• Extremum points: If f 0 (x) > 0 a function f is increasing at x, if it is negative it
is decreasing, if f 0 (x) = 0 then x is a critical point. If f 00 (x) > 0 at a critical point
it is a minimum, if f 00 (x) < 0 at a critical point it is a maximum, if f 00 (x) = 0
it is undecided, it could be a point of inflection, a point joining a concave up
f 00 (x) > 0 interval from a concave down, f 00 (x) < 0, interval.
• Extrema: A point x = a is an relative or local maximum if there is an open interval
I containing a such that f (a) ≥ f (x) for all x in I, a point x = a is an relative or
local minimum if there is an open interval I containing a such that f (a) ≤ f (x) for
all x in I.
• Absolute extrema: A point x = a is an absolute or global maximum if f (a) ≥ f (x)
for all x in the domain, it is an absolute or global minimum if f (a) ≤ f (x) for all x
in the domain.
• Newton’s method to find f (x) = 0 start with an initial guess x1 and iterate using
xn+1 = xn −
f (xn )
f 0 (xn )
(1)
until it converges or is obviously not working.
• The anti-derivative and the indefinite integral: For a function f (x) the function
F (x) is the anti-derivative if
dF (x)
= f (x)
(2)
dx
The indefinite integral is the family of all anti-derivatives
Z
f (x)dx = F (x) + C
(3)
where C is the arbitrary constant of integration.
• Integration table
1
Stefan Sint, sint@maths.tcd.ie, see also http://www.maths.tcd.ie/~sint/MA1S11.html
1
R
f (x)
f (x)dx
n+1
xn (n 6= −1) xn+1 + C
sin x
− cos x + C
cos x
sin x + C
• Linearity: if
Z
Z
f (x)dx = F (x) + C,
g(x)dx = G(x) + C
(4)
then
Z
Z
λf (x)dx = λF (x) + C,
[f (x) + g(x)]dx = F (x) + G(x) + C
(5)
where λ is a constant.
Questions
1. (2) Find the absolute maximum of
1
f (x) = − x4 + x3 + 2x2 + 1
4
2. (2) Use Newton’s method to find
as a starting value.
√
5
(6)
3 to within 0.01 by solving x5 = 3, with x1 = 2
3. (4) Find the indefinite integral of x3 − 4x,
√
√
√
x + 1/ x, x3 − 3 x and x(1 + x4 ).
Extra Questions
The questions are extra; you don’t need to do them in the tutorial class.
1. Evaluate
Z
1
dx
(7)
1 − sin x
Hint: first multiply by 1 + sin x above and below the line, then use the identity
cos2 x + sin2 x = 1.
2. Find the points of inflection of
f (x) = x4 − 6x2 + 12x + 24
2
(8)