Evaluate: 4cos(2x) dx = 2 sin(2x) = 0.
- B This is not a related-rate problem; this is a differential equation! It just happens to involve a
rate.
Step 1: If we translate the first sentence into an equation we get = kR.
Put all of the terms that contain an R on the left of the equals sign and all of the terms that
contain a t on the right-hand side.
= k dt
Step 2: Integrate both sides.
∫ = k^ dt
Step 3: If we solve this for R, we get R = Cekt (see the unit on Differential Equations).
Now we need to solve for C and k. First, we solve for C by plugging in the information that the
radius is 4 initially. This means that R = 4 when t = 0.
If 4 = Ce^0 , then C = 4
Next, we solve for k by plugging in the information that R = 10 when t = 2.
10 = 4e^2 k
= e^2 k
ln = 2k
ln = k
Step 4: Now we have our final equation: R = 4.
If we plug in t = 3 we get R = 4 ≈ 15.81.
- A We use the Second Fundamental Theorem of Calculus to find the derivative of the integral. In
