Fundamental Theorem of Calculus
The integral of f'(x) is...
f(x) + C
What is the antiderivative of cos?
sin
What is the integral from 0 to 5 of x2 + 1?
46 and 2/3
Consider the definite integral from 0 to 4 of x2. Would a right-Riemann sum be an over or underestimate?
Over.
A pump allows water to flow into a pool at the rate of r(t) liters per minute, where t is the time in minutes since the pump was turned on. Write an integral which gives the total amount of water accumulated in the tank between 2 and 3 minutes.
int(2 to 3) rt dt
What is the antiderivative of x2?
x3/3
What is the area under the curve of the graph of sin2(x)cos(x) between x = 0 and x = pi?
0
Simplify:
d/dx ( int (from 2 to x) of sin(t4)dt ) = [ ]
sin(x4)
What app was TikTok's predecessor based on?
Vine
A particle's speed is given by v(x), where v(x) is the particle's velocity relative to the origin at x seconds. Which function gives the total distance traveled by the particle after s seconds?
D(s) = int(0 to s) v(x) dx
Take the antiderivative of xcos(x2+4)dx
u - sub w/ u = x2 + 4, so du/2 = xdx, so 1/2 * int(cos(u)du) = 1/2 sin(u) + C = 1/2 sin(x2+4) + C
Mr. Smith's speed in miles per hour on a racetrack is given by -2x2+100x+40. How far did he go in 3 minutes?
9.2 miles. (It's a fast car.)
Let g(t) = sin(t). Let f(x) = int(0 to x) of sin(t)dt , on the interval x from 0 to 2pi. On what interval(s) is f(x) decreasing?
From pi to 2pi.
What is the most-watched TikTok video?
A 10-second blank screen.
State, roughly, in your own words, the Fundamental Theorem of Calculus.
If F(x) = int(a to x) f(t)dt, then F'(x) = f(x).
Or, int(a to b) of f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
Take the antiderivative of 1/(-x2+6x-10) dx
-arctan(x-3)+C
Fill in the blank: int(a to b) of f(x) dx = [ ]int(b to a) of f(x) dx.
negative
Let g(t) = sin(t). Let f(x) = int(0 to x) of sin(t)dt , on the interval x from 0 to 2pi. On what interval(s) is f(x) concave down?
pi/2 to pi and 3pi/2 to 2pi.
How many people visit TikTok in a month?
1.5 to 2 billion.
Let f be a differentiable function such that f(3) = 2.345 and f'(x) = ln(x2+1). What is f(5)?
FTC says that
int(3 to 5) of f'(x) dx = f(5) - f(3), hence f(5) = int(3 to 5) of f'(x) dx + f(3), which by the calculator is 7.976.
Take the antiderivative of (x3+2x2+3x+4)/(x+1)
x3/3 + x2/2 + 2x + 2ln|x+1| + C
Fill in the blanks: int (a to c) of f(x) dx = int (a to b) of f(x) dx [ ] int ([ ] to c) of f(x) dx
+, b
Use a left-Riemann sum to approximate the area under the curve of -x2+16 from x = 0 to x = 4. Have four rectangles.
f(0)+f(1)+f(2)+f(3) = 50
What does TikTok's Chinese name translate to in English?
"vibrating sound"