Math Worksheets

Question: Need AP Calculus Help!!!!?

I have a worksheet due in math and I can’t do a few of them. Can anyone please help?

1) The Kertz Leasing Company leases fleets of new cars to large corporations. The rental fee is $2000 per car per year. However, for contracts with a fleet size of more than 10 cars, the rental fee per car is discounted by 1% for each car in the contract, up to a maximum fleet of 75 cars. How many cars leased to a single corporation in one year will produce (A) maximum revenue and (B) maximum profit if each car depreciates in value $1000 per year.

and….

2) A point P is taken on the curve y=x^3. The tangent at P meets the curve again at Q. Prove that the slope of the curve at Q is four times the slope at P.

Answer: 2) Let (p, p^3) and (q, q^3) be the two points on the curve. Since dy/dx = 3x^2, the tangent at point P has a slope of 3p^2. Likewise, the slope of the tangent at Q is 3q^2.

The slope of the line passing through (p, p^3) and (q, q^3) is
(p^3 – q^3) / (p-q), which is
(p-q)(p^2 + pq + q^2) / (p-q) =
p^2 + pq + q^2

This is equal to the slope of the tangent at point P, since it’s the same line. So
p^2 + pq + q^2 = 3p^2
-2p^2 + pq + q^2 = 0
2p^2 – pq – q^2 = 0
(2p + q)(p – q) = 0
So either p-q=0 or 2p+q = 0, meaning either p=q or p = (-1/2)q. We know p and q are not the same, because that would Mean points P and Q are the same. So p = (-1/2)q

The slope of the tangent at Q is 3q^2. The slope of the line at p is 3p^2, which is 3(q^2 / 4). This shows that the slope at Q is four times as big as the slope at P.

Finding the Slope Given 2 Points