graph of f′(x) that is positive from x = 2 to x = ∞. The graph in (D) satisfies all of these
requirements.- C Remember that ln(u(x)) = .
We will need to use the Chain Rule to find the derivative.f′(x) = (3) = − 3tan(3x)- B The Second Fundamental Theorem of Calculus tells us how to find the derivative of an
integral. It says that f(t) dt = f(u) , where c is a constant and u is a function of x.Here we can use the theorem to get = .Now we evaluate the expression at x = −4. We get = 2.- D Velocity is the first derivative of position with respect to time.
The first derivative is: v(t) = 2t − 7.Thus, the velocity of the particle is zero at time t = 3.5 seconds.- A We can use u-substitution to evaluate the integral.
Let u = sin^2 x and du = 2 sin x cos x dx. Next, recall from trigonometry that 2 sin x cos x =sin(2x). Now we can substitute into the integral ∫eu du, leaving out the limits of integration for
the moment.Evaluate the integral to get ∫eu du = eu.
Now, we substitute back to get esin(^2) x
.
Finally, we evaluate at the limits of integration to get
- B In order to find the average value, we use the Mean Value Theorem for Integrals, which says