Four linked problems drawn from this topic’s subtopics. 12 parts · untimed · answers and worked solutions at the end.
Example 1
A production model has marginal cost C′(x)=ax+b dollars per unit, for 0≤x≤6. The fixed cost is C(0)=50 dollars. The total costs are C(2)=206 and C(4)=386 dollars. Both a and b are unknown.
Find a.
Show worked answer
Answer: 6
Integrate: C(x)=2ax2+bx+50.
The two totals give 156=24a+2b and 336=8a+4b.
Subtract twice the first equation from the second: 24=4a.
Therefore a=6.
Example 2
Consider f(x)=3x2+0 and g(x)=9x+0, for x≥0. The graphs meet at x=0 and one other point, enclosing a finite region.
a Find the x−coordinate of the other intersection.
Find the area of the enclosed region.
y
01230102030
x
f(x)=3x2+0g(x)=9x+0Show worked answer
Answer: 13.5
On 0<x<3,g is the upper curve, and the positive gap is 3x(3−x).
Area =∫03(3x(3−x))dx.
An antiderivative is 3(23x2−3x3).
The area is 3×627=13.5 square units.
Example 3
Consider f(x)=−3x2−3 and g(x)=−27x−3, for x≥0. The graphs meet at x=0 and one other point, enclosing a finite region.
a Find the x−coordinate of the other intersection.
b Find the area of the enclosed region.
A vertical line x=t cuts off three quarters of the enclosed area to its left. Find t in 0<t<9. Give your answer to 3 decimal places.
y
0369-200-1000
x
f(x)=−3x2−3g(x)=−27x−3Show worked answer
Answer: 6.063
The area to the left of t is 3(29t2−3t3).
Set this equal to 0.75×364.5. Dividing by the full area and putting u=9t gives 3u2−2u3=0.75.
On 0<u<1, the derivative 6u(1−u) is positive, so there is exactly one solution in this interval.
Numerically u=0.6736481777, hence t=6.062833599.
Therefore the answer is 6.063 to 3 decimal places.
Applications of Integration Review Practice | Year 12 Methods | Ascent Arcade