2 - chapter 12 - further application of calculus

Question 1

convex curve

Answer 1

curves upwards
second derivative >0

Question 2

concave curve

Answer 2

curves downwards
second derivative <0

Question 3

if second deriv = 0

Answer 3

point of inflection - curve goes from concave to convex or vice versa
f’(x) is not necessarily = 0 - doesn’t have to be a horizontal point of inflection

Question 4

if f’(x) = 0 and f’‘(x) = 0

Answer 4

a point can be any one of the 3 stationary points - to see which one look at the grad each side
see table in textbook p251

Question 5

cartesian eq

Answer 5

just involves x and y

Question 6

parametric eq

Answer 6

involve a 3rd variable (parameter)
eg x = f(t) and y = g(t)

Question 7

differentiating parametric eq

Answer 7

dy/dt = dy/dx *dx/dt
dy/dx = (dy/dt) / (dx/dt)
or = dy/dt * dt/dx

Question 8

tangent paralel to x and y

Answer 8

tangent parallel to x axis
dy/dt = 0
tangent parallel to y axis
dx/dt = 0

Question 9

integrating parametric eq

Answer 9

∫y dt * dx/dt
= ∫y dx
if between limits eg x = a and x = b have to find the t values when x = a and b to integrate y with those limits and with respect to t

Question 10

related rates of change

Answer 10

eg dV/ dr = dV /dt * dt/dr

Question 11

area between 2 curves

Answer 11

y = f(x) and y = g(x)
A = ∫f(x) - g(x) from b to a
where f(x) is above g(x)

Question 12

area between curve and y axis

Answer 12

y = f(x)
rearrange to be x = g(y)
and integrate g(y) within the limits