Chapter 3 Review

If f prime of c = 0, then f has a local maximum or minimum at c

false, there could be a point of inflection

How do you find critical numbers of a function?

find the first derivative of the function and find where the function equals zero by setting the function equal to zero and solving.

What is a local maximum of a function?

the largest y-value of an interval of a function.

What is an absolute maximum of a function?

the largest y-value of the function

the curve y = f of x (f(x)) has the horizontal asymptote y = L means...

(the larger x becomes the closer it gets to L??)

it is extremely important to remember to add what to the end of the antiderivative?

+C (or D, E, F, etc.)

what is the antiderivative of sin x?

-cos x

To evaluate the limit at infinity of any rational function, we first must....

...divide both the top and bottom by the highest power of "x" in the bottom of the rational function.

What are the 3 hypotheses that need to be satisfied in order for Rolle's Theorem to work?

1 - the function must be continuous on the closed interval [a, b] 2 - the function is differentiable on the open interval (a, b) 3 - the function's value at "a" equals the function's value at "b" ( f(a)=f(b) ) (the graph starts and ends and the same y-value)

If you have a graphing calculator or computer, why do you need calculus to graph a function?

Because calculus gives exact values to specific points of the graph without trial and error

What are the steps when sketching a graph?

Domain, Intercepts, Symmetry, Asymptotes, Intervals of Increase/Decrease, Local Max/Min, Concavity/Inflection points, then sketch

Explain how the Closed Interval Method works.

First, ensure the function is continuous within the closed interval. Second, find the values of the critical numbers & endpoints of the function within the interval. Third, identify the largest and smallest values to identify the ABSOLUTE max and min respectively.

How do you prove a function has exactly one real root/solution?

First, show that a real root exists via the Intermediate Value Theorem. Then use Rolle's Theorem and argue by contradiction that if the function had two roots, f(a)=0=f(b) but if the DERIVATIVE of the function is always increasing/decreasing, it can never be 0, therefore it can't have two real roots.

What is Fermat's Theorem?

If a function has a local maximum or minimum at a number (c) & if the derivative of the function at that number (c) exists, then the derivative of that number equals zero or the slope of the tangent line at that number is 0.

What is Rolle's Theorem?

If the three needed hypotheses are satisfied, there is a number (c) in an open interval (a, b) such that the derivative at that point equals 0 or the slope of the tangent line at that point equals 0.

What is needed before utilizing the Mean Value Theorem and why?

Rolle's Theorem because the first and second needed hypotheses in Rolle's theorem also apply to the MVT.

How to determine which line is the function, first derivative, or second derivative?

Start by looking at the x-intercepts. If there is a corresponding local max or min at every x-intercept you have identified the first derivative because it is the intercepting line while the local max/min is the function. Inflection points of the first derivative will correlate with the local max/min of the second derivative.

What is an inflection point on a graph?

This is a point on the graph where the concavity changes/the tangent line overlaps the function's line.

What is the Second Derivative Test?

Using the critical numbers (c) obtained from the first derivative, where f prime of x = 0, evaluate the second derivative at these critical numbers. If the second derivative evaluated at a critical number is greater than 0, the original function has a local minimum there. The opposite is true if it is a local max.

What is the general formula to find the next iteration using Newton's Method?

Xn+1 = Xn - f(Xn)/f'(Xn)

What is an antiderivative of a function?

a function whose derivative is the known function.

What is the First Derivative Test?

after identifying the critical numbers of a continuous function, where the first derivative = 0, determine the sign behavior before & after the critical number(s). If f prime goes from a positive to negative value, the critical number is a local max of the function. If f prime goes from a negative to a positive value, the critical number is a local min of the function.

What are the exceptions to Fermat's Theorem?

horizontal tangent lines, or where tangent lines are not possible, e.g. vertical tangent lines & corners

Simply describe the Intermediate Value Theorem.

if a function is continuous on a closed interval [a, b] and a value of the function doesn't equal another value of the function, say if f of a = -1 and f of b = 5, there is a horizontal line the function must intersect between those values.

What is the Extreme Value Theorem?

if a function is continuous on a closed interval [a, b], then there is an absolute max and min within that interval.

What is the Mean Value Theorem?

if a function is continuous on the closed interval [a, b] & differentiable on the open interval (a, b), then there is a number (c) in the open interval where its differentiable value is equal to the function's value at "b" minus the functions value at "a" all over the value "b" minus the value "a".

What is the Decreasing Test?

if the derivative of a function is negative on an interval, the function is decreasing on that interval.

What is the Increasing Test?

if the derivative of a function is positive on an interval, the function is increasing on that interval.

What does it mean if a function is concave up on an interval?

if the graph of a function is always above the tangent lines, it is concave up. If the second derivative is positive it is concave up.

f double prime of x = 0 means?

inflection point on the function

What does it mean if f double prime of x is greater than 0 (f''(x)>0) for x<2 and f double prime of x is less than 0 (f''(x)<0) for x>2?

it means there is an inflection point at x=2 (2, f(2)) and it is concave up from negative infinity to 2 and concave down from 2 to infinity. It also infers there could be a local minimum before x=2 and a local max after x=2

When finding the antiderivative of a function, can a power to a variable be negative one?

no, because the method to find the antiderivative of powers requires n + 1 in the denominator which a -1 makes it undefined

if f prime of x (f(x)) equals zero, the tangent line is....

parallel to the x-axis

What is the antiderivative of sec x tan x?

sec x

what is the antiderivative of cos x?

sin x

what is the antiderivative of sec^2 x?

tan x

What does it mean if f prime (f'(x)) doesn't change sign before and after a critical number when doing the First Derivative Test?

that the tangent line at the critical number is parallel to the x-axis but will continue to have the same direction after the critical point.

What is a critical number of a function?

the critical number(s) of a function is (are) a number (c) in the function's domain such that the derivative at that number equals zero or does not exist.

What does it mean when the limit of f of x as x approaches infinity equals L?

the values of f of x can be made arbitrarily close to L by requiring x to be sufficiently large OR the limit of f of x, as x approaches infinity, OR f of x approaches L as x approaches infinity

What is the Mean Value Theorem in layman's terms?

there is at least one point P(c, f(c)) on the graph which has a tangent line with a slope equal to the slope of the secant line of the end points of the closed interval.

When is the line y = L called a horizontal asymptote?

when the limit of f of x is L as x approaches infinity or negative infinity

What are you finding when setting the first derivative of a function equal to zero?

where the graph of a function is is a local max or min because at those points, the tangent will have a slope of 0/be parallel to the x-axis.

How to find x and y intercepts?

x intercept: set equation equal to zero y intercept: evaluate equation at zero

The limit of 1 over x to the r power as x approaches infinity or negative infinity is equal to what?

zero