- C There are two ways we could evaluate this limit. First, we could recognize that this limit is in
the form of the Definition of the Derivative . Then the limit is the
derivative of f(x) = tan x at x = . The derivative of f(x) = tan x is f′(x) = sec^2 x, and at x = ,
we get f′ = sec^2 = 2. Second, we have a limit of the indeterminate form , so we can
use L’Hôpital’s Rule to find the limit. Take the derivative of the numerator and the
denominator: = . Now we take the limit to get sec^2
= 2.
- A First, rewrite the integral as .
Now, we can simplify the integral to .
Next, use the power rule for integrals, which is ∫ xn dx = + C.
Then, we get = .
- A The simplest thing to do here is to find of both pieces of the function and set them equal to
each other. We can do this by plugging x = 1 into both pieces: 1 − 3k + 2 = 5 − k. If we solve
for k, we get k = −1.
- C The curve y = 8 + 2x − x^2 is an upside-down parabola and looks like the following:
