Derivatives
The derivative of y = 3x2 - 4
dy/dx = 6x
What does the first derivative represent at a specific point?
The instantaneous rate of change of a function.
A bacterial population is modeled by P(t) = 100 + 20t. What is P′(t)?
P'(t)=20 bacteria per unit of time.
How many bones does a typical adult human have?
206
The first derivative of f(x) = -6x2 + 10x - 14.
f'(x) = -12x + 10
The process of finding critical points.
Setting first derivative equal to zero.
The height of a plant is H(t) = t² + 2t. Find its instantaneous growth rate at t = 3.
H'(t)=2t+2, so H'(3)=8 units of height per unit of time.
What color is an airplane's black box usually?
Bright orange
The first derivative of y = sqrt(x)
y' = 1/ (2sqrt(x) )
The condition for a function f(x) to be increasing.
What is f'(x) > 0
A population follows P(t) = −t² + 12t. At what time is its growth rate zero?
t=6, because P'(t)=-2t+12=0.
What is the chemical symbol for gold?
Au
The first derivative of
y = 6x^4 - 1/2 x^2 - 10
y' = 24x^3 - x
If f′(x) = 0 at a point, what does this tell us about the tangent line?
The tangent line is horizontal because its slope is zero.
An enzyme reaction rate is modeled by v(T) = −(T − 37)² + 100. At what temperature is the rate maximal?
T=37, where the reaction rate reaches its maximum of 100.
Which TV series features the fictional paper company Dunder Mifflin?
The Office
Find the derivative of
(10x^2-7x)/(3x^4-5x^6)
f'(x) =
((20x-7)(3x^4-5x^6))/((12x^3-30x^5)(10x^2-7x))
Can a function be continuous at a point but not differentiable there? Give an example.
Yes. For example, \(f(x)=|x|\) is continuous at \(x=0\), but it is not differentiable there because the graph has a sharp corner.
A cell culture has biomass B(t) = −2t² + 16t + 5. Find the time when biomass reaches its maximum and calculate that maximum.
The maximum occurs at t=4, and the maximum biomass is B(4)=37 biomass units.
In The Matrix, which color pill does Neo take to learn the truth about his reality?
The red pill