Question

Let F of x equals the integral from 1 to 3 times x of the natural logarithm of t squared. Use your calculator to find F″(1).

A. 12
B. 6
C. 4
D. 1/9


Find the range of the function f of x equals the integral from 0 to x of the square root of the quantity 16 minus t squared.

A. [−4, 4]
B. [0, 4]
C. [0, 4π]
D. [0, 8π]

Answer 1

Not sure whether you mean f(t) = ln(t^2) or f(t) = (ln t)^2
In either case, F'(x) = f(3x) * 3
then take the next derivative

#2. You can do the math, recalling arcsines, etc, or just realize that the integral represents the area under the circle of radius 2, in the first quadrant. As you increase x from 0 to 4, the area increases from 0 to 1/4 of a circle, or 4π.