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.