Chapter 3 Flashcards
What are the two ways to analyze trends or slopes?
difference quotient
differential quotient
What is the difference quotient?
used to determine the average slope between two points by using secant
difference quotient=slope=secant
What is a curve
graph of nonlinear function
compromised of parabolas and hyperbola
What is a secant?
line connecting TWO points
gives the rate of change
it intersects with the curve
e.g.
f(x2)-f(x1)/x2-x1
What is f(x) in an exponential called?
the production function
What is the differential quotient
the slope of entire curve by calculating the tangent
the tangent only touches one point of the curve to measure the slope
What is a tangent
line of contact of ONE POINT
just touches a point, doesn’t intersect with it (unlike secant) and can be determined by using a slope triangle
you can calculate approximately by using the limit of the secant line
what does the limit formula mean?
when two points are close and they are hardly indistinguishable and the difference is a very small number it can be expressed by using limit.
What does the △ triangle symbol in the limit formula mean?
the change
What does lim mean?
△x->0
limit value of the change tends to be zero.
What is the differential?
used to calculate approximately changes in the function value f(x) with respect to changes in the independent variable x if it changes by dx units
e.g.
dy=f’(x)dx
What are the different forms of a derivative function?
f’(x)
df(x)/ dx or d/dx f(x)
dy/dx
What does the first derivative mean and what does the second derivative mean?
Or the higher order derivatives
1st order derivative: indicates the slope of the curve of f(x)
in business: it is also called limit/marginal function because it indicates approximate change by unit
2nd order derivative: indicates how the slope of the curve of f(x) changes
if the curve gets steeper or flatter
How do you make the notation that a derivative is positive?
f’(x) = ax>0
basically the function is bigger than zero
How do you make a note that function is negative?
f’(x) = ax<0
function is smaller than zero
What happens to the derivative if x=0
We cannot calculate slope, the function is undefined
What happens if you put a positive number for f’(x)
the derivative is also positive
What does this mean
∀ ?
For all
If the function is increasing and convex, what are the first and second order derivatives?
f’(x) >0
f”(x)>0
If the function is decreasing and convex, what are the first and second order derivatives?
f’(x)<0
f”(x)>0
If the function is decreasing and concave, what are the first and second order derivatives?
f’(x) <0
f”(x)<0
If the function is increasing and concave, what are the first and second order derivatives?
f’(x) >0
f”(x)<0
What is an inflection point?
It is where the curve changes
It can be calculated using second derivative when x=0
But you also have to meet the condition that the third derivative is positive and not zero!!! -to prove that it’s an inflection point
What is the slope of the curve of the cost function called?
cost curve
What is Diminishing Returns?
After a certain level of input, adding an additional input leads to smaller increases in output and can eventually cause output to decrease.
What other areas of business can you use the first derivative?
-marginal consumption & savings rates
-marginal rate of substitution (labor vs capital, good A vs good B)
-elasticity of supply and demand
-marginal utility
