Sequences and Series
T(n+1) = T(n) + 5, T(1) = 8
Give the simplified rule to find the nth term.
T(n) = 3 + 5n
Differentiate
y = -4t^7 + t^4/12 + 5t
y' = -28t^6+t^3/3 + 5
Anti-differentiate
y' = 12x^2 - 5
y = 4x^3 - 5x + c
Find the velocity formula if
x(t) = t^2 - 6t
v(t) = 2t - 6
T(n+1) = 4T(n), T(1) = 2
Give the general rule in the format
T(n) = ar^n
T(n) = 1/2 * 4^n
Differentiate
y = (3x^3 - 9x^4)/(6x^2
y' = 1/2 - 3x
Find f(x) if
f'(x) = (x - 2)(x + 2)
f(x) = x^3/3 - 4x + c
For
x(t) =t^3 - 6t^2 + 9t + 2
Find the initial displacement and velocity.
x = 2 m
v = 9m/s
Calc Allowed
For the G.P. if T(13) = 12288 and T(16) = 98304
Find T(17) and T(1).
T(17) = 196608
T(1) = 3
Determine the coordinates of the points on the curve
y = -2x^3
where the gradient is -24.
(2,-16), (-2,16)
Find the Antiderivative of
f'(x) = 2x - 3x^2
if f(2) = 0
f(x) = x^2 - x^3 + 4
The displacement of a particle from the origin O at time t seconds is x where
x(t) = 12 + 4t - t^2
Find the speed of this particle at 5 seconds.
6 m/s
For the following two geometric series, state why or why not S(infinity) is possible.
a) 120+ 90 + 67.5 + ....
b) 64 + 96 + 144 + ....
a) Yes, r = 0.75
b) No, r = 1.5
For the function
f(x) = 2x^2 + 5x + 6,
find the coordinate of the point that has a tangent parallel to the x-axis.
(-5/4, 23/8)
After major repairs to the floor of a swimming pool, it is refilled at a flow rate of 500(t + 2) Litres/h.
Calculate the instantaneous flow rate when the volume of water in the pool is 35000L.
6000 L/h
A body moves in a straight line in accordance with the displacement-time function
x(t) = 2t^2 - 18t + 30
Find the distance travelled in the first 7 seconds.
142 metres
Find x given that 8 + 11 + 14 + ... + x = 1020
x = 77
Consider the equation
f(x) = x^3 - 2x^2 + 5
Find the gradient function using differentiation from first principles.
f'(x) = 3x^2-4x
No Calculator
Given that
f'(x) = a/x^2 - 2
where f'(3) = 4, f(3) = -4, find f(6)
f(6) = -1
The velocity of an object is given by
v(t) = (t - 2/t)^2
where t represents the time (seconds) after the object passes through the origin. The displacement, x (metres), of the object 3 seconds after passing through the origin is -3 m. Determine the equation for the displacement of the object.
x(t) = t^3/3 - 4t - 4/t + 4/3