Unit 3

Wild card

100

If f(x)=ln(3x), then f′(1) is

1

200

Let f(x)=a sin(3x), where 𝑎 is constant. 𝑓′(𝜋)=2, then 𝑎 is equal to

-2/3

300

Without using a calculator, find all the values of x𝑥 between 0 and 2𝜋 for 2cos(3x)+√3=0

5π/18, 7π/18, 17π/18, 19π/18, 29π/18, 31π/18

400

Solve each of the following equations for x:

ln(x^2−2x+8)=2lnx

x=4

500

Given that logaN=0.5(loga24−loga0.375−6loga3), find the value of 𝑁.

8/27

Integration

100

If dy/dx=1/x^2 and y=2 when x=1, then y=

y=−1/x+3

200

∫x^2−(1/x^2)+sinx. dx   is   

(x^3/3)+(1/x)−cosx+c

300

The area bounded by the curve y=1/(3−x), the 𝑥-axis, the 𝑦-axis and the line 𝑥=2 is

ln (3) 

400

∫sinx/cosx   .dx between pi/4 and pi/3

1/2ln2

500

Evaluate ∫e^−(x/10)sin(2x)    .dx between 0 and pi, correct to four decimal places.

0.1345

Derivatives

100

If y=(x^4+x)/x^2, then 𝑑𝑦/𝑑𝑥 equals

2x−1/x^2

200

If y=(4−9x^4)^1/2, then dy/dx = 

-18x^3(4-9x^4)^-1/2

300

The graph of y=ax^3+bx^2+cx+d touches the line 2y+6x=15 at the point A(0,7.5) and has a stationary point at B(3,−6). Find the values of 𝑎, 𝑏, 𝑐 and 𝑑.

a=2/3,  b=−2.5,  c=−3,  d=7.5

400

Find the coordinates of the stationary points of the curve with equation y=x/(x^2+1)

(1,0.5),  (−1,−0.5)

500

A culture contains 1000 bacteria and 5 hours later the number has increased to 10000. The number, 𝑁, of bacteria present at any time, t𝑡 hours, is given by 𝑁=𝐴𝑒^(𝑘𝑡).

Find this rate of growth when t=4 

dN/dt≈2905.7

Logs

100

If log2(8x)+log2(2x)=6, then x=

2

200

Determine the equation of the asymptote for the function f(x)= log9(x-3)-4

x=3

300

solve for x - log4(4x+16)-log4(x^2-2)=1

x=3, -2

400

Solve the equation log5x=16logx5

x=625 or x=1/625

500

It is conjectured that the area affected by an earthquake, 𝐴 km^2, is related to the magnitude of the earthquake on the Richter scale, 𝑅, by the formula

R=2.3log10(A+4800)−7.5for 1≤R≤8

Determine the area affected by an earthquake of magnitude 7.

2010537.68 km

Wild card

100

The gradient of the normal to the curve y=e^(−cosx) at the point where 𝑥=𝜋/3 is

−2e^(1/2)/√3

200

Rainwater is being collected in a water tank. The rate of change of volume, V L, with respect to time, 𝑡 seconds, is given by

dV/dt=5t+2

The volume of water that is collected in the tank between times 𝑡=2 and 𝑡=6 is

88 L

300

A long trough with a parabolic cross-section is 1.5 metres wide at the top and 2 metres deep. Find the depth of water when the trough is half full.

1.26 m

400

After 𝑡 seconds (𝑡≥0), a particle has acceleration 𝑎=𝑘𝑡 m/s^2, where 𝑘 is a constant. Given that, after 2 seconds, the particle’s displacement is 7 m and its velocity is 4 m/s, find the value of 𝑘.

3/8

500

Rewrite the equation y=3ln(x)−4 with 𝑥 as the subject.

x=e^(y+4)/3