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7.03 Finding f(x) from f'(x)

Worksheet
Primitive functions
1

For each of the following find the equation of y:

a
\dfrac{d y}{d x} = 4 x + 7 and y passes through the point \left(3, 41\right).
b
\dfrac{d y}{d x} = 9 x^{2} - 10 x + 2 and y passes through the point \left(2, 13\right).
c
\dfrac{d y}{d x} = 9 x^{\frac{2}{3}} and y passes through the point \left(8, \dfrac{889}{5}\right).
d
\dfrac{d y}{d x} = \sqrt{ 6 x + 66} and y passes through the point \left( - 5 , 17\right).
e
\dfrac{dy}{dx} = \left( 4 x - 2\right)^{3} and y passes through the point \left(2, 79\right).
2

Find the equation of the curve that has a gradient of 15 x^{2} + 7 and passes through the point \left(2, 59\right).

3

Find the equation of y in terms of x given that \dfrac{d y}{d x} = 12 \left(x - 1\right)^{3} and y = 247 when x = - 2.

4

Find the equation of the curve that has a gradient of \dfrac{d y}{d x} = \left(x - 6\right)^{2} and the point \left(3, - 4 \right) lies on the curve.

Applications
5

The velocity v \left( t \right), in \text{m/s}, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t, \text{ where } t \geq 0The object is initially at the origin.

a

Find the displacement x \left( t \right)= \int v(t) \, dt of the particle at time t.

b

Find the time at which x(t) = 54 \text{ m}.

6

The velocity v, in metres, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 6 t + 10, \text{ where } t \geq 0

a

Find the displacement s \left( t \right) of the particle at time t, given that the object starts its movement at 8 \text{ m} to the right of the origin.

b

Find the displacement of the object after 5 seconds.

7

The velocity v \left( t \right), in metres, of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t^{2} + 30 t + 9, \text{ where } t \geq 0The object starts its movement at 6 \text{ m} to the left of the origin.

a

Find the displacement, s \left( t \right), of the particle at time t.

b

Find the displacement of the object after 5 seconds.

8

The velocity of a particle moving in rectilinear motion is given by v \left( t \right) = 4 t - 4 where v is the velocity in metres per second from the origin and t is the time in seconds. The particle is instantaneously stationary when it is 1 \text{ m} right of the origin.

a

Find the time when the particle is stationary.

b

Determine the function x \left( t \right) for the position of the particle.

c

Find the value of x(t) at:

i

t=0

ii

t=1

iii

t=2

9

The velocity v \left( t \right) of an object travelling horizontally along a straight line after t seconds is modelled by v \left( t \right) = 12 t^{2} - 48 t, \text{ where } t \geq 0The object starts its movement 5 \text{ m} to the right of the origin.

a

Find the displacement, x \left( t \right), of the particle at time t.

b

Find the times, t, when the object is at rest.

c

Find the displacement at which the object is stationary other than its initial position.

10

The velocity v \left( t \right), in \text{ m/s}, of an object along a straight line after t seconds is modelled by v \left( t \right) = 12 \sqrt{t}The object is initially 5 \text{ m} to the right of the origin.

a

Find the function x \left( t \right) for the position of the particle.

b

Hence, calculate the position of the object after 9 seconds.

11

The acceleration a, in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by a \left( t \right) = 6 t - 27, \text{ where } t \geq 0After 10 seconds, the object is moving at 90 \text{ m/s} in the positive direction.

a

State the velocity, v, of the particle at time t.

b

Find all the times at which the particle is at rest.

12

The acceleration a, in \text{ m/s}^{2}, of an object travelling horizontally along a straight line after t seconds is modelled by a = 2, \text{ where } t \geq 0The object is initially 8 \text{ m} to the right of the origin and moving to the left at 5 \text{ m/s}.

a

Find the velocity, v, of the particle at time t.

b

Find the displacement, x, of the particle at time t.

c

Find the position of the object at 7 seconds.

d

Find the time at which the particle is moving at a speed of 3 \text{ m/s} to the right.

13

In a closed habitat, the population of kangaroos P \left( t \right) is known to increase according to the function P' \left( t \right) = \dfrac{t}{2} + 9 where t is measured in months since counting began.

a

Calculate the total change in the population of kangaroos in the first 4 months since counting began.

b

Find the number of months it will take from when counting began for the population of kangaroos to increase by 88.

14

The rate of emission \dfrac{d E}{d t} of CFCs in Australia, measured in tonnes per year, was given by \dfrac{d E}{d t} = 100 + \left(\dfrac{60}{1 + t}\right)^{2}where t is the time in years after 1 September 1988.

a

Find the rate of emission on 1 September 1988.

b

Find the rate of emission on 1 September 1997.

c

Find the value that \dfrac{d E}{d t} approaches as the years pass by.

d

Calculate the total amount of CFCs emitted in Australia during the years 1988 to 1997.

15

The total revenue, R (in thousands of dollars), from producing and selling a new product, t weeks after its launch, is given by \dfrac{d R}{d t} = 401 + \dfrac{500}{\left(t + 1\right)^{3}}

a

Given that the initial revenue at the time of launch was zero, state the revenue function.

b

Find the average revenue earned over the first 5 weeks.

c

Calculate the revenue earned in the 6th week.

16

An ice cube with a side length of 25 \text{ cm} is removed from the freezer and starts to melt at a rate of 25 \text{ cm}^{3}/\text{min}. Let V be its volume t minutes after it is removed from the freezer.

a

State the equation for the rate of change of volume.

b

State the equation for the volume, V, of the cube as a function of time t.

c

Find the value of t when the ice cube has melted completely.

17

M \left( x \right) is the cost of producing x units of a certain product, where M \rq \left( x \right) = 9 x^{2} - 70 x + 500.

a

Determine the cost function M \left( x \right) in terms of x.

b

Hence, determine the extra cost incurred by producing 60 units rather than 20 units.

18

Water flows into, then out of, a container at a rate \dfrac{d V}{d t} litres per minute given by \dfrac{d V}{d t} = t \left(10 - t\right) where the number of minutes, t \geq 0.

a

Sketch the graph of \dfrac{d V}{d t}.

b

Hence, find the maximum flow rate.

c

Find an expression for the volume of water, V litres, in the container at time t minutes, assuming that the container is initially empty.

d

Find the total time taken for the container to fill and then empty.

19

The mass, m grams, of a raindrop falling for t seconds is increasing at a rate given by \dfrac{d m}{d t} = \dfrac{1}{120} \left(\sqrt{t} + \dfrac{t^{2}}{12}\right) \text{ g/s}

a

Given that the initial mass of the raindrop is zero, determine the mass m of the raindrop after t seconds.

b

Determine the exact mass of the raindrop after 16 seconds have passed.

c

Another raindrop starts as a gas particle with a mass of 0.004 \text{ g}. How much heavier will it be after 16 seconds than a raindrop that is initially weightless?

20

Wheat is poured from a silo into a truck at a rate of \dfrac{d M}{d t} , where \dfrac{d M}{d t} = 81 t - t^{3} \text{ kg/s} and t is the time in seconds after the wheat begins to flow.

a

Find an expression for the mass M \text{ kg} of wheat in the truck after t seconds, if initially there was 1 tonne of wheat in the truck.

b

Calculate the total mass of wheat in the truck after 8 seconds.

c

Find the largest value of t for which the expression for \dfrac{d M}{d t} is physically possible.

21

A pen moves along the x-axis ruling a line. The graph of the velocity of the tip of the pen as a function of time is shown below. The velocity, in \text{cm/s}, is given by the equation v = 4 t^{3} - 36 t^{2} + 72 twhere t is the time in seconds. The tip is initially 5 \text{ cm} to the right of the origin.

a

Find an expression for x, the position of the tip of the pen, as a function of time.

b

What feature will the graph of x as a function of t have at the point where t = 3?

c

What happens to the pen at t = 3?

d

The pen uses 0.02 \text{ mg} of ink per centimetre travelled. How much ink is used between t = 0 and t = 4? Round your answer to two decimal places.

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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-7

applies the concepts and techniques of indefinite and definite integrals in the solution of problems

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