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9.01 Integrating exponential functions

Worksheet
Exponential functions
1

Find the primitive function of the following:

a
e^{ 2 x}
b
e^{ - 5 x }
c
e^{ 0.25 x}
d
9 e^{ 3 x}
e
e^{x} + e^{ - 3 x }
f
e^{ 3 x} - e^{ - 8 x }
g
\dfrac{1}{6} \left(e^{ 0.5 x} + e^{ 3 x}\right)
2

Find the following indefinite integrals:

a
\int e^{ 2 x} dx
b
\int e^{ - x } dx
c
\int e^{ \frac{1}{2} x} dx
d
\int 4 e^{ 2 x} dx
e
\int e^{4 - 3 x} dx
f
\int e^{3 - 2 x} dx
g
\int 4 e^{1 - 2 x} dx
h
\int \left(e^{ 0.5 x} + e^{ 3 x}\right) dx
i
\int \left(e^{t} - 5\right) dt
3

Find the exact value of the following definite integrals:

a
\int_{ - 4 }^{2} e^{x} dx
b
\int_{0}^{3} e^{ 4 x} dx
c
\int_{0}^{2} e^{ \frac{3}{2} x} dx
d
\int_{ - 2 }^{4} e^{ 3 x} dx
e
\int_{ - 3 }^{5} e^{ - 2 x } dx
f
\int_{0}^{2} \left(7 + e^{x}\right) dx
g
\int_{0}^{3} \left( 2 e^{ 5 x} + e^{ - 7 x }\right) dx
h
\int_{ - 3 }^{3} \left(e^{ \frac{1}{4} \theta} - 3 e^{ - 0.2 \theta }\right) d\theta
4

Find the primitive function of the following:

a
4^{ x}
b
3^{ 2x}
c
2^{ 5x+1}
d
-2\times3^{ -x}
5

Consider the function y = x^{2} e^{x}.

a

Find an expression for \dfrac{dy}{dx}.

b

Hence find \int x \left( 2+ x \right) e^{x} dx.

Area under a curve
6

Find the exact area enclosed by:

a

The curve y = e^{ 4 x}, the x-axis, and the lines x=1 and x=2.

b

The curve y = e^{x}, the x-axis, the y-axis, and the line x = 2.

c

The curve y = -e^{x}, the x-axis, the y-axis, and the line x = 5.

d

The curve y = e^{x} + 4, the coordinate axes, and the line x = 4.

e

The curve y = e^{ - 6 x}, the x-axis, and the lines x = - 2 and x = 1.

f

The curve y = 6 e^{ 3 x}, the x-axis and the lines x = - 3 and x = 3.

g

The curve y = -2^{ x}, the x-axis, and the lines x=1 and x=2.

7

Consider the curve y = e^{x} - 9.

a

Find the exact value of the x-intercept.

b

Find the exact area between the curve y = e^{x} - 9 and the x-axis, from the lines x = 0 to x = 3.

8

Consider the curve y = e^{x}.

a

Find the area bound by the curve, the x-axis, the y-axis, and the line x = 3.

b

Find the equation of the tangent to the curve y = e^{x} at the point where x = 3.

c

Find the exact area enclosed between y = e^{x}, the x-axis, the y-axis, and the tangent at x = 3.

Applications
9

Find the equation of the curve given the gradient function and a point on the curve:

a

\dfrac{d y}{d x} = e^{ 3 x}, and point \left(0, -1\right).

b

\dfrac{d y}{d x} = e^{ 2 x}, and point \left(0, 6 \right).

c

\dfrac{d y}{d x} = e^{ x} -2x, and point \left(0, 4 \right).

d

f' \left( x \right) = {x^2}e^{2x^3}, and point \left(0, 0\right).

10

A particle P starts off from a fixed point O, with an initial velocity, v, of 2\text{ m/s}. Its acceleration a \text{ m/s}^2 after t seconds is given by:

a = e^{ - t }
a

Find v\left(4 \right), the velocity of P after 4 seconds, correct to two decimal places.

b

Find the displacement, x, of the particle P after 4 seconds, correct to the nearest hundredth of a metre.

11

An object is cooling and its rate of change of temperature, after t minutes, is given by: T' = - 10 e^{ - \frac{t}{5} }

a

Find the instantaneous rate of change of the temperature after 8 minutes, to two decimal places.

b

Find the total change of temperature after 9 minutes, to two decimal places.

c

Hence, find the average change in temperature over the first 9 minutes, to two decimal places.

12

Fluid leaks from a storage facility and its rate of change after t days is given by:

F' \left( t \right) = 3000 e^{ - 0.5 t}
a

Find the amount of fluid that will leak in the first 7 days, correct to two decimal places.

b

Find the amount of fluid that will leak in the next 7 days, correct to two decimal places.

13

A particle's acceleration a, is measured in\text{ cm/s}^2 according to the equation:

a = - e^{ 4 t}
a

If the particle is initially at rest, find an equation for the velocity, v, of the particle at time t.

b

Find the change in the particle's position after 5 seconds, to the nearest centimetre.

14

The initial velocity of a particle is 10\text{ m/s}, and its acceleration in \text{ m/s}^2 is given by:

a \left( t \right) = 2 e^{ 3 t}
a

Find its velocity, v \left( t \right), after t = 2 seconds. Round your answer to the nearest metre per second.

b

Let x \left( t \right) be the displacement of the particle at t seconds. If its initial position is 2 \text{ m} to the left of the origin, find its displacement after 4 seconds, correct to the nearest metre.

15

A particle moves so that its velocity over time t, in metres per second, is given by:

v = 2 e^{t} - 1

If the particle's displacement, x, is 10 when t = 0, find its exact displacement when t = 3.

16

A rechargeable battery has a voltage of 9 \text{ V} when fully charged. When the battery is used to run an electronic toy, the voltage, V volts, remains at 9 \text{ V} for 30 minutes and then it decreases instantaneously, at a rate modelled by:

\dfrac{d V}{d t} = - 0.3 e^{ - 0.03 t}
a

Find the change in the battery voltage after the toy has been in use for 45 minutes. Round your answer to two decimal places.

b

Find the function V, the battery voltage after the toy has been in use for t minutes when t \geq 30.

c

Find the number of whole minutes the battery can be used to run this toy, if a minimum voltage of 7 \text{ V} is required.

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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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