Advanced Placement Calculus Answers
- Question #1: The function given by:
is differentiable at x=0. What is the value of f(n-m)?
Solution: Since f(x) is differentiable at x = 1, it must be continuous there, which means that
f(0) = n+1
n + 1 = 4, and n = 3
Now we can determine m from the fact that f(x) is differentiable at x = 0.
The right-hand branch slope is
and f'(0) = 2.
The left-hand branch slope is f'(0) = m.
m = 2
At this point we can calculate n – m = 1
f(1) = 3+e2
- Question #2: If y = sin(x3), what is dy/dx?
u(x) = x3
du/dx = 3x2
y(u) = sin(u)
dy/du = cos(u)
dy/dx = (dy/du)(du/dx) = cos(x3)·3x2
dy/dx = 3x2cos(x3)
- Question #3:
- Question #4: What is the value of a, if:
(c) 1 + ¶
(e) 1 + e
Answer: u = 2 + sin(ax)
du/dx = a·cos(ax)
ln(3/2) = (1/a)ln(3/2) and a = 1.
- Question #5: Which of the following are antiderivatives of f(x) = 2x?
(a) 2x/ln(2) + ln(2)
(b) 22x/ln(2) + ln(2)
(c) x2/ln(2) + 1/ln(2)
(d) x + 2ln(2)
(e) 2x + ln(2)
The easiest way to solve this problem is to calculate the derivative of all 5 possible answers. We find that answer (a) is correct:
d(2x/ln(2) + ln(2))/dx = ln(2)2x/ln(2) + 0 =2x