Derivatives
f(x)= ln x
f'(x) = ?
1/x
∫(ex) dx=?
ex +c
lim(x→2) (3x+1)
7
What theorem is this:
if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b) then there exists a point c where the function's derivative at c equals its average rate of change over [a,b]
Mean Value Theorem (MVT)
The velocity of a car is modeled by the equation: v(t) = 2t3 + 7t + 4
Find its acceleration at t = 0.
7
f(x)= cos x
f'(x)= ?
-sin x
f(x)= ∫0cos x (t2) dt
f'(x) = ?
-sin x cos2 x
lim(x→1) (x2 - 1)/(x-1)
2
Let g be a continuous function on the closed interval [-3,3] where g(-3) = 0 and g(3) = 6.
What theorem guarantees a y-value of 5 from [-3,3]?
Intermediate Value Theorem (IVT)
A tank fills at the rate R(t) = 2t4 - t2 - 4, measured in liters per hour. Find R'(t) at time t = 2 hours and explain the meaning of that value in the context of this problem.
60 - The rate of the rate in which the tank fills at hour 2 is 60 liters per hour2.
f(x) = x4 + 3x3 - 4x2 + 13
f'''(x) = ?
f'''(x) = 24x + 18
∫ (4x6 -2x3 + 7x -4) dx
4/7(x7) - 1/2(x4)+ 7/2(x2) -4x + c
f(x)={
x2+1 if x<2
3x−1if if x≥2
lim(x→2) f(x)
5
What theorem states: if g(x) ≤ f(x) ≤ h(x) and if lim (x→a) g(x) = L and lim (x→a) h(x) = L then lim (x→a) f(x) = L
Squeeze Theorem
The rate of change of radius r of a circle is 4 cm/s. Find the rate of change of Area A when r=2cm.
16π
f(x)= ln |ex|+7x
f'(x)=?
8
∫6x2 (x3 + 4)5 dx
1/3 (x3 + 4)6 + c
lim(x→0) (xsin(x)) / (x2)
1
What theorem is this?
If f(x) = ∫ax from a to x f(t)dt and f is continuous, then f'(x) = f(x)
Fundamental Theorem of Calculus
A tank is being filled w/ water at a variable rate. The rate (in gallons per minute) at which water flows into the tank is given by the function R(t) = 4 + sin((πt)/6) where t is the time in minutes, for 0 < t < 6.
How much water flows into the tank during the first 6 minutes?
24 + (12/π) gallons
f(x)= arctan (4x)
f'(x) = ?
4/(1+16x2)
∫(x√(x2 + 1)) dx
1/3 * (x2 + 1)3/2 + C
lim (x → 2) x3 - 7x2 + 10x / x2 + x -6
-6/5
What Theorem is this:
If a function is continuous on a closed interval, [a,b], then it attains a max and min value on that interval
Extreme Value Theorem
A ball is dropped from a height and its height above the ground is given by h(t) = (5t2 -20)/(t-2) for t doesn't = 2
Find the height of the ball as it approaches 2 seconds
20