Given f(x) = , then f′(x) =
Here we have f′(t) = .
Next plug in t = −1 and solve.
9. A You should know that ∫ = ln |x| + C.
We take the antiderivative and we get dx = 4 ln |x − 1| + C.
Next, plug in e + 1 and 2 for x, and take the difference: 4 ln(e) − 4 ln(1).
You should know that lne = 1 and ln 1 = 0. Thus, we get 4 ln(e) − 4 ln(1) = 4.
- C We find the total distance traveled by finding the area of the region between the curve and the
x-axis. Normally, we would have to integrate but here we can find the area of the region easily
because it consists of geometric objects whose areas are simple to calculate.
The area of the region between t = 0 and t = 4 can be found by calculating the area of a triangle
with a base of 4 and a height of 60. The area is (4)(60) = 120.
The area of the region between t = 4 and t = 8 can be found by calculating the area of a
rectangle with a base of 4 and a height of 30.
The area is (4)(30) = 120.
The area of the region between t = 8 and t = 16 can be found by calculating the area of a
trapezoid with bases of 4 and 8, and a height of 90 (or you could break it up into a rectangle
and a triangle). The area is (4 + 8)(90) = 540.
Thus, the total distance traveled is 120 + 120 + 540 = 780 kilometers.
