2 - 8 - applications of calculus

Question 1

maclaurin series

Answer 1

f(0) + f’(0)x + f’‘(0)/2! x² +… + f^r(0)/r! x^r

Question 2

deriving maclaurin series

Answer 2

any series (that can keep being differentiated) can be written as
f(x) = a0 + a1x + a2x² + a3x³+
then differentiate to find f’, f’’ etc
then use f(0), f’(0), f’‘(0) etc to find a0, a1, 2a2, 3!a3 etc
substituting back in gets the series

Question 3

improper integrals - ∫ f(x) from infinity to a

Answer 3

have to integrate from b to a then tend b to infinity
(if the limit exists and if finite)
if the limit is infinite you say the integral diverges

Question 4

improper integrals - undefined point in the range

Answer 4

if f(x) is not defined at x = k
then to integrate f(x) from c to a
∫ f(x) from b to a and tend b to k
∫f(x) from c to b and tend b to k
if the limit is infinite you say the integral diverges

Question 5

volume of revolution around x axis

Answer 5

V = 𝞹 ∫y²

Question 6

volume of revolution around y axis

Answer 6

V = 𝞹 ∫x²

Question 7

volume of revolution between g(x) and f(x)

Answer 7

V = 𝞹 ∫(g(x)² - f(x)²)

Question 8

parametric volumes of revolution

Answer 8

x = f(t) and y = g(t)
about x axis:
V = 𝞹 ∫y² dx/dt dt
about y axis:
V = 𝞹 ∫x² dx/dt dt

Question 9

mean value of f(x) from a to b

Answer 9

1/(b-a) ∫f(x) dx from b to a