Seven linked problems across the course · 21 parts · 60 minutes. Work on paper; answers and full working at the end.
Example 1
Let f(x)=1+Ae−kx10 for all real x. The constants satisfy A>0 and k=0.
Given f(0)=35, determine A.
Show worked answer
Answer: 5
- At zero, f(0)=1+A10=35.
- Rearranging gives 1+A=6 and A=5.
Example 2
An unknown number n of parcels is dispatched independently, each with the same unknown probability p of arriving late. The number X arriving late has E(X)=2.2 and Var(X)=1.76. A reserve team can handle at most m late parcels.
a Find n.
Find the smallest m for which P(X≤m)≥0.8.
Show worked answer
Answer: 3
- Use X ~ Bin(11,0.2).
- P(X≤2)=0.6174015488<0.8, whereas P(X≤3)=0.8388608≥0.8.
- Therefore the smallest capacity is m=3.
Example 3
Let f(x)=a(x−3)ln(x−3) on x>3. The constant a is nonzero.
a Given f′(3+e)=−8, determine a.
b Find the x−intercept of the tangent at x=3+e2. Give your answer to 3 decimal places.
How many distinct real solutions does f(x)=−2 have in its domain?
Show worked answer
Answer: 1
- Divide by the nonzero constant a=−4 and put t=x−3>0. The equation becomes tlnt=0.5.
- The function tlnt decreases to its minimum −e1 at t=e1, then increases without bound. As t→0+, it approaches 0 from below.
- A positive target meets only the increasing branch after t=1, so there is one solution.