Autumn 2024 Flashcards
When is a function defined as increasing?
A function is called increasing if x1<x2 implies that f(x1) ≤ f(x2)
When is a function defined as decreasing?
A function is called decreasing if x1<x2 implies that f(x1) ≥ f(x2)
When is a function defined as strictly increasing?
A function is called strictly increasing if x1<x2 implies that f(x1) < f(x2)
When is a function defined as strictly decreasing?
A function is called strictly decreasing if x1<x2 implies that f(x1) > f(x2)
What do graphs of linear functions look like?
Graphs of linear functions are straight lines
What general formula are power functions defined by?
Defined by f(x) = Ax^r where x > 0 and both A and r are constants
What determines the shape of the graph of power functions?
The value of r (the power) determines the shape of the graph of power functions
What do demand functions tell us?
Demand functions tell us how the quantity demanded depends on the price
What formula is used to calculate the average rate of change from a to a+h?
f(a + h) − f(a) / h
What formula notates the instantaneous rate of change of f at a?
f’(a)
What formula is used to calculate the relative rate of change (percentage change) of f at a?
f’(a)/f(a)
How can we tell if a function is differentiable at a point x?
The function f is differentiable at a point x if the following limit exists:
f’(x) = lim h–> 0 f(x+h) - f(x) / h
What do you get if you differentiate loga(x)?
loga(x) = ln(x)/ln(a) so f’(x) = 1/xln(a)
State the rule given by logarithmic differentiation
If f(x) = ln h(x) then f’(x) = h’(x) / h(x)
What do you get if you multiply a function by its inverse function?
g{f(x)} = x
State the differentiation rule for a function and its inverse
g’(x){f(x)} = 1/f’(x)
State the formula for finding the linear approximation to a function
The linear approximation to f about x=x1 is:
f(x) ≈ f(x1) + f’(x1)(x-x1) when x-x1 is small
What is the differential of y=f(x)?
The differential of y=f(x) is given by dy = f’(x) dx
How do we denote the percentage change in y for each percentage change in x?
Δy/y / Δx/x
State the formula used to calculate the elasticity of a function y=f(x) at x1
f’(x1)*x1/f(x1)
When is a function that has been created by combining other functions continuous?
Any function that can be constructed from continuous functions by combining one or more of addition, subtraction, multiplication and division except by zero and excluding composition, is continuous at all points where it is defined
Does continuity imply differentiability?
Continuity does not imply differentiability
If f is differentiable at x=a…
f is continuous at x=a
What is an extreme point?
An extreme point is either a maximum or a minimum point
When is x* a maximum point?
xεD is a maximum point of f if f(x)≥f(x) for all xεD
When is x* a minimum point?
xεD is a minimum point of f if f(x)≤f(x) for all xεD
When is f(x*) the extremum?
f(x*) is the extremum if it is either the maximum or minimum value of the function
What is the relationship between interior extreme points and the tangent line?
At interior extreme points, the tangent line is horizontal
If x* is an interior extreme point…
f’(x*)=0
