Unit 3 revision PPT for EHS study lessons

1. Solve:
1.
(a) log5 125 = 𝑥 (b) log𝑥 81 = 4
(c) log𝑒(2𝑥 − 1) = −3 (d) ln 𝑥 + 5 − ln 3𝑥 = 0
(e) log6 𝑥 − 3 + log6(𝑥 + 2) = 1
2.
(a) 𝑒3𝑥 − 9 = 0 (b) 𝑒2𝑥 − 5𝑒𝑥 = 0
(c) 𝑒2𝑥 + 𝑒𝑥 − 2 = 0

2. 3. Sketch 𝑦 = log𝑒 𝑥 − 5 , showing x-intercept and the asymptote.

3. 4. Determine the derivative of the following:
(a) 𝑓 𝑥 = ln 3𝑥 − 1
(b) 𝑓 𝑥 = 4𝑒2𝑥−1
+ cos(5𝑥)
(c) 𝑓 𝑥 =
2
(5−2𝑥)
(d) 𝑓 𝑥 = 𝑥3 sin(2𝑥)
(e) 𝑓 𝑥 =
𝑒2𝑥
sin(𝑥)

5. Determine the indefinite integrals of the following:
(a) 3𝑥2
+ 8𝑒4𝑥
𝑑𝑥
(b) (5𝑥 − 3)5 𝑑𝑥
(c)
3
(3−2𝑥)
dx

6. 1. Determine the exact value of the following definite integrals (give your solutions in simplest form):
(a) 0
2
𝑒2𝑥
− 2𝑥 𝑑𝑥
(b) 0
2𝜋
3 2 sin 2𝑥 𝑑𝑥
2. Determine the equation of the curve that passes through the point (0,3) if the gradient is given by
𝑑𝑦
𝑑𝑥
= 2𝑒2𝑥
+ 𝑒−𝑥
.
3. The gradient function of a particular curve is given by f ′(x)=cos(2x)−sin(2x). Determine the rule for this
function if it is known that the curve passes through the point (π,2).

7. 1. Differentiate 𝑥𝑙𝑛(𝑥) and hence determine an antiderivative of ln(𝑥).
2. Differentiate 𝑦 = 2𝑥𝑒3𝑥 and hence determine an antiderivative of 𝑥𝑒3𝑥.
3. Use your result from above to determine 0
1
𝑥𝑒3𝑥 .

8. (TA) Calculate the approximate area under the curve 𝑓 𝑥 = ln(𝑥2
+ 1) between 0 and 2 with 4 strips. Give
your answer to 4 decimal places.
(a) Using the trapezoidal rule.
(b) Using a graphics calculator evaluate the reasonableness of the answer.
(TA)

9. TF
1. A curve is represented by the equation 𝑦 = 𝑎𝑥 cos 3𝑥 where a is a constant. If
𝑑𝑦
𝑑𝑥
= −5 when 𝑥 = 𝜋, what is the
value of a?
2. A particle moves in a straight line so that its displacement a point, O, at any time, t, is 𝑥 = 3𝑡2 + 4.
Determine the velocity and acceleration when t=2.
3. The acceleration of a particle moving horizontally in a straight line can be expressed as 𝑎 = 2 + 6𝑡 𝑚/𝑠2.
(a) If initially the velocity is 3 m/s, determine the equation that expresses velocity as a function of time.
(b) Calculate the displacement when t = 2 seconds.

10. TA
The apparent brightness, B of a star can be found using the formula 𝐵 = 6 − 2.5 log 𝐴 , where A
is the actual brightness of that star.
(a) Determine the apparent brightness of a star with actual brightness of 3.16.
(b) Determine the actual brightness of a star with apparent brightness of 8.

11. Determine the area of the pink shaded region.