calc unit 3

Derivatives at a point

• find derivative

• Plug in x or y values

Equations of tangent and normal lines

• Tangent line

  • find derivative (FINDS SLOPE)

  • Then find the POINT by plugging in x value to ORIGINAL equation

  • Put in point slope form

    • y - y1 = m(x-x1)

• Normal line 

  • Same steps

  • The slope will be the opposite sign and reciprocal of the tangent line equation

Horizontal Tangent Lines

• Find derivative

• Find when the derivative equals zero

  • These are your x-values

• Plug in the x values found into the original equation

Vertical Tangent Lines

• Find derivative

• Find when the derivative is undefined (set equal zero)

Horizontal Tangent Lines Implicitly 

• Find derivative implicitly 

• Set the numerator equal to zero

• Find both x and y

• Plug into original equation 

  • Eliminate any answers that do not work

Vertical Tangent Lines Implicitly 

• Find when denominator is 0 

• Solve for y in terms of x

• Plug into original equation 

• Plug the x values found into the original equation to make sure it is valid 

Derivative Motion

• Things to know (position functions)

  • s(t) - location of an object in relation to a point or origin

  • Derivative of s(t) is velocity or v(t)

  • Derivative of v(t) is acceleration or a(t)

• Speed

  • Speed is the absolute value of velocity

  • Speed is increasing if v(t) and a(t) are both the same sign

  • Speed is decreasing if v(t) and a(t) are different signs

  • A positive v(t): direction is right or up

  • A negative v(t): direction is left or down

• Position

  • Set the function equal to the height and factor

  • Choose the value that matches the equation

  • Plug into the derivative (or whatever it is asking)

• Average velocity

  • Displacement/change in time

    • f(b)-f(a)/b-a

Graphing Derivatives with Original Function

• X- intercepts

  • Where the derivative equals zero (x-intercept) that is where the original functions local max and mins are

• Slopes

  • The function has a positive slope, the derivative is positive (increasing)

  • The function has a negative slope, the derivative is negative (decreasing)

• Concavity

  • Points of inflection for the functions are the derivatives local max and mins (THE SLOPE OF THOSE POINTS OF INFLECTION IS WHERE YOU PLACE THEM)

Absolute Extreme Values 

• Where an y value is larger or smaller than all y values on the domain

• WORDING

  • Where they occur is x-value

  • Absolute max or min is the y-value

• Infinity is not an absolute max or min

• Extreme Value Theorem

  • If the function is continuous on an interval, there is guaranteed a max and min

Local Extreme Values 

• Largest or smallest value on SOME interval

Solving for max and min

• A function can only have max or min when the FIRST DERIVATIVE is undefined or equal to zero

• Steps

  • Find derivative

  • See where it becomes undefined or zero (critical value)

    • This is where you set the denominator to zero (for undefined)

    • This is where you set the numerator to zero (for equaling zero)

  • Check endpoints 

    • Plug in the domain to the original function

  • Find the smallest and largest

• Finding domain

  • Set the denominator not equal to zero (more than and less than)

  • Find x 

• Piecewise functions for finding max and mins

  • Find derivative of each piece separately 

  • Check for continuity with double sided limit

  • Then check if it is undefined

    • Plug in the value where the piecewise function equals on both functions

    • If they do not equal, then the other critical value is the one that the function equals in the piecewise

  • Check limits to infinity to check if one is absolute

• Absolute value

  • Prove that the absolute value is positive from stating that x is more than zero

    • This gets rid of the absolute value

  • Solve for critical points like normal but remember restriction on domain

  • Solve for both situations (where x could be negative)

    • Solve like normal (with different restriction)

  • Then plug into original equation

Increasing and decreasing

• Find critical value 

• Make sign chart

  • determine the interval according to the sign chart

  • The x values are the local max or min depending on the direction of sign chart

  • If the signs do not change, no local extrema

• First derivative test

  • Find derivative

  • Find where it is undefined or equals zero

  • Use those x values to make a sign chart

    • If signs change there are local extrema

  • Plug in the x-values that have extrema into original equation

• How to tell if there is a absolute extrema there as well

  • Use limits to find end behavior

Concavity

• Concave up

  • Second derivative is positive

• Concave down

  • Second derivative is negative

• Steps

  • Find second derivative

  • Find when it is undefined or zero

  • Use sign chart and write intervals

• Points of inflection

  • This is when the concavity changes

  • Find the second derivative

    • Find undefined or when equal to zero

    • Sign chart

  • Plug into original equation (x-values that switch)

  • Put in (a,b) format

Second derivative test

• When the second derivative is positive, the local maximum at the x value found

• When the second derivative is negative, the local minimum is the x-value found

• Steps:

  • Find first derivative

    • Find x-values at 0 

  • Find second derivative and plug in the x-values

Graphing

• Requires original function

  • X and y intercepts

  • Asymptotes

• Requires first derivative

  • Increasing and decreasing

  • Max and min

• Requires second derivative

  • Concavity

  • Points of inflection

Local Linearity

• Given (a,f[a])

• Linearization equation: y= f(a) + f’(a)[x-a]

• Set up

  • Make the function in terms of x

  • Let x = number

  • Make the variable “a” something easy to solve

• Writing to solve

  • Original function = f(number) ~ L(number) = linearization equation

Differentials

• Using tangent lines to approximate the value of the function

• Dy = f’(x) times dx

• Set up

  • Dx is the difference between the two positions

  • Replace x with the first position

• Implicitly

  • Solve like implicit differentiation

• Word problems

  • take equation for what you are trying to solve

  • manipulate into linearization equation

Newton’s Method

• x(n) + 1 =  x(n) - f(xn)/f’(xn)

• Set up

  • State continuity 

  • Choose two values that create the root by IVT

  • Choose a guess inbetween those intervals

  • Plug into newton's method equation

  • Hit enter on calculator till two values match

Tangent and Secant Approximations

• Concave up

  • Tangent underestimates true value

  • Secant overestimate true value

• Concave down

  • Tangent overestimate true value

  • Secant understimates true value

• Set up tangent line

  • Find derivative

  • Plug in x value given to both the derivative and regular

    • These give you your slope and second y value

  • Put into point slope form

  • Plug in the approx value into x

  • Proving

    • Find second derivative to see concavity (pos - up, neg - down)

• Set up secant line

  • Use the domain given to find two ordered pairs

    • Find the slope of them

  • Plug into the y = mx + b at the approx value

Rolles Theorem

• If both f(x) are equal to each other

  • This means that the derivative equals 0 for some “c” on interval

• Set derivative equal to 0 and solve

Mean Value Theorem for Derivatives

• [a,b]

• F’c = f(b)-f(a)/b-a

• Set up

  • Find derivative of function in terms of c and set equal to the MVT value to find c

L’hopitals rule

• The limit as x approaches a of f(x)/g(x) is equal to the limit as x approaches a of the derivative of f’(x)/g’(x)

• REMEMBER: state limits equal zero INDIVIDUALLY

• Dealing with infinites

  • Simplify your trig functions