Theorems, Rules, and Tests

When is a function continuous?

1. f(c) is defined

2. limit f(x) as x→c exists

3. limit f(x) as x→c = f(c)

What is a removable discontinuity? What is a non-removable discontinuity?

hole → change one place to make it continuous; VA, jump asymptote, etc. → have to change more than one place to make it continuous

What is the Intermediate Value Theorem (IVT)?

If f(x) is continuous on the interval [a, b] with f(a) ≠ f(b), then if d is any number between f(a) and f(b), there is at least one c between a and b such that f(c) = d

What is the Extreme Value Theorem (EVT)?

If f(x) is continuous on the interval [a, b], then f has both an absolute maximum value and an absolute minimum value on [a, b]

What are the best steps to finding a limit?

1. direct substitution

2. If you get an indeterminate value form

   1. factoring and simplifying

   2. rationalizing the number

   3. combine fractions

   4. use a graph/ table of values

   5. trig identities/ limits

What is the limit of (sinx)/x when x→0 AND direct substitution results in 0/0

1

What is the limit of (1 - cosx)/x when x→0

0

What is a derivative?

the slope of the tangent line of a graph at a particular point (a tangent line only touches the graph at one point)

How do you find the derivative?

1. f’(x) = limit of [(f(x + h) - f(x)) / h] as h→0

2. limit of [(f(x) - f(c)) / (x - c)]

How is a function being continuous related to the function being differentiable?

If a function is differentiable at x=c, then is is continuous at x=c; If a function is not continuous at x=c, then it is not differentiable at x=c

Does a function have to be differentiable for it to be continuous?

no → cusp, absolute value graph, etc.

How do you know if a function is differentiable at a given point?

if the derivative from the left = the derivative from the right

What is the Constant Rule?

d/dx [c] = 0 (the slope of a horizontal line is 0)

What is the Power Rule?

d/dx [x^n] = n * x^(n-1)

What is the Constant Multiple Rule?

d/dx [c * f(x)] = c * f’(x)

What is the Sum and Difference Rule?

d/dx [f(x) ± g(x)] = f’(x) ± g’(x)

What is the Product Rule?

d/dx [f(x) g(x)] = [f(x) * g’(x)] + [g(x) * f’(x)]

What is the Quotient Rule?

d/dx [f(x) / g(x)] = ([g(x) * f’(x)] - [f(x) * g’(x)])/ [g(x)] ^ 2

What are the steps to implicitly differentiate a function?

1. differentiate both sides with respect to x → (dy/dx)

2. collect all terms with dy/dx on one side

3. factor out all dy/dx

4. solve for dy/dx

What are related rates?

the use of the chain rule to find the rates of change of 2 or more variables that are changing with respect to time

What are the steps/ guidelines to solve a related rates problems?

1. Make a sketch and label all the given quantities and all the quantities you must find

2. Write an equation involving the variables whose rates of change are either given or to be determined

3. Find the derivative with respect to time

4. Substitute and solve the equation

What is optimization?

the process of finding an absolute maximum or an absolute minimum for a specific quantity

What are the steps to optimization?

1. Make a sketch

2. Write a primary equation for the quantity to be optimized

3. Reduce the equation to one independent variable if possible

4. Find the optimum value by substituting in known values or making another equation to find other values
   

What is d/dx [sinx]

cosx

What is d/dx [cosx]

-sinx

What is d/dx [tanx]

sec²(x)

What is d/dx [cotx]

-csc²x

What is d/dx [secx]

secx * tanx

What is d/dx [cscx]

-cscx * cotx

What are critical numbers?

when f’(c) = 0 or is undefined; is a possible relative extrema

Where do absolute extrema occur?

critical numbers or endpoints

How do you find (absolute) extrema?

1. Find the critical numbers of f(x)

2. Evaluate the original function at each of the critical numbers

3. Evaluate the original function at each endpoint of the interval

4. Whichever of the y-values is the greatest and the least are your absolute extrema

What is Rolle’s Theorem?

If f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists some value c in (a, b) such that f’(c) = 0

What is the Mean Value Theorem (MVT)?

If f is continuous on [a, b] and differentiable on (a, b), then there is some value c in (a, b) such that f(c) = the slope of the secant line that passes through a and b

What is The First Derivative Test?

1. Find critical numbers of f(x)

2. Make a sign chart of f’(x)

3. Use the sign chart to decide if f is increasing/ decreasing/ constant

4. If f’(x) changes from positive to negative it’s a relative maximum; If f’(x) changes from negative to positive it’s a relative minimum

What is The Second Derivative Test?

1. Find the critical numbers of f(x)

2. Find f’’(x) at each of those points

3. If f’’(c) < 0 it’s a maximum, f’’(c) > 0 it’s a minimum, if f(c) = 0 do the First Derivative Test

How do you know if a function is concave up or concave down?

1. If f’’(x) > 0 on (a, b) → concave up

2. If f’’(x) < 0 on (a, b) → concave down

What are inflection points? Where do they occur?

when f(x) changes from concave up to concave down; f’’(x) = 0 or undefined