AP Calculus - Basic Derivatives (Chap. 2)
Justify: That a maximum occurs on f(x) and x = b
A maximum occurs at x = b if b is a critical point and if the slope of f changes from + to - at b. So, f '(x) changes from positive to negative at x = b
Justify: That a minimum occurs on f(x) and x = b
A minimum occurs at x = b if b is a critical point and if the slope of f changes from - to + at b. So, f '(x) changes from negative to positive at x = b
Def: Given a position function x(t) = ___..... what is a "Stationary Point."
A stationary point is t = ___, when x ' (t) = 0. This special type of a critical point is called "stationary" since the derivative of position (velocity) is negative.
Def: What is a critical point on f(x)?
Any point x = c where f '(c) = 0, called a stationary point, or where f '(c) = undefined. So, if c is part of the domain, a critical point is where the derivative is 0 (horz. tangent) or where the derivative is undef. (vert. tangent).
Take the antiderivative of f(x)=1/x or x^(-1)
By definition, and proved later with through implicit differentiation. Absolute value is required since you start with the domain of 1/x (all neg. and pos #) to a more restirced domain with ln(x) (only pos #).
Def: Derivative (d/dx) of logarithmic functions: y=log₃(x)
Change of base leads to this derivative
Memory trick: Derivatives/Antiderivatives of sin(x) and cos(x)
Clockwise gives derivative while counterclockwise gives antiderivative
Take the derivative (d/dx) of f(x)=1/x or x^-1
Don't get tricked! Power rule works fine for the derivative.
Ex: Take the derivative (d/dx) of y = x⁴
Example of the power rule
Justify: Where does the maximum or minimum occur on a function f(x) given the closed interval [a,b]?
Extreme Value Theorem: An absolute (or "global") max or min can only occur at the endpoints of the interval or at a critical point on f(x), x = c. You must compare f(a), f(b), and f(c)
Initial Value Problem (IVP). Find the particular solution to the DE x '(t) = 8 t+5 if x(2)=23
First, a general solution is found, then the initial value t = 2 and x(2)=23 are substituted in order to solve for the constant c.
How does Horizontal Scaling in a function affect its derivative? (d/dx) y = (3x)^5
Horizontal scaling will also appear in the derivative as well as a vertical scaling. The horizontal scaling by 1/3 causes a vertical scaling of 3. Later, this is called the chain rule.
How does Horizontal Translation in a function affect its derivative? (d/dx) y = (x+5)^9
Horizontal translation in a function also appears in its derivative. The shift of 5 to the left appears graphically in both f(x) and f ' (x)
Justify: Is x = p a point of inflection on f(x)?
If the concavity of f(x) changes sign at x = p: Test: POI occurs at x = p if: 1) p is part of the function's domain 2) f '' (p) = 0 or Undef. 3) f '' changes sign at x = p
Ex: Derivative (d/dx) of y = ln(x)
No absolute value needed (moving from a more restricted domain to a less restricted domain)
Differential Equation ex) Is y = sin(5x) a solution to the second order DE y'' = 5y ?
No, since the second derivative does not equal 5 times the first derivative. y'' = 5y' but y'' does not equal 5y.
Take the antiderivative of a power function f(x) = 3x^5
Reverse the power rule.
Take the derivative (d/dx) of the cubic root of x to the 5. f(x)=³√x⁵
Rewrite exponent as a fractional power then use power rule.
Take the antiderivative of an exponential function f(x) = 2^x
Since the derivative of 2 to the x is itself times a number (ln2), then the antiderivative is itself divided by a number (ln2).
What is the average rate of change (ARC) of f(x) between x=a and x=b
The average rate of change is simply the slope between two points (the difference quotient)
Def: Derivative of Exponential Functions. For example d/dx) y = 7^x
The derivative of an exponential is itself, times a number (vertical scaling of ln(b))
Derivative of a constant function: f(x) = 5 so find f ' (x)
The derivative of any constant function is 0. Remember the derivative is the slope and this graph is horizontal with no slope.
Find the general solution to the Diff. Eq. x '(t) = 8 t+5
The general solution to a DE is the family of functions from which the DE was derived which always includes the process of taking the antiderivative and the solution including the + c
Instantaneous rate of change (IRC) of f(x) (Definition of a Derivative Version 1)
The slope of f(x) at any point x
Instantaneous rate of change (IRC) of f(x) at the point x=a (Definition of a Derivative Version 2)
The slope of f(x) at point x=a
How does Vertical Scaling in a function affect its derivative? (d/dx) y = 8x^6
Vertical scaling in a function also appears in its derivative. The stretching of 8 vertically in f(x) while cause f ' (x) to stretch vertically as well.
How does Vertical Translation in a function affect its derivative? (d/dx) y = 4x^5+100
Vertical translation has no effect. The shift of 100 vertically in f(x) will not affect f ' (x) graphically.