Calculus I - Chapter 3

This distance traveled problem and the area problem are

the same problem

lim 1/x x-> infinity

0

What is the local and absolute max/ min values of cosine?

1 and -1

Steps to determine the intervals where the function is increasing or decreasing

1. Find the derivative 2. Find the critical points 3. Find where the derivative is positive or negative. To do this we use the critical points. We graph them on a number line and pick test points to see if the derivatives are positive and negative. Make sure you chose points in the DERIVATIVE not the function

What are the steps to find the absolute max/minimum (The Closed Interval Method)

1. Find the values (the y values) of the critical numbers 2. Find the values at the end points 3. The largest of the values if the absolute max and the smallest is the absolute minimum value.

Steps for finding the critical numbers

1. Simplify expression if needed and find the derivative 2. Once you find the derivative, simplify again if needed because we will need to derivative to pretty fairly easy to solve for. 3. To find when the derivative equals zero, set the numerator equal to zero and solve for x. 4. Then check where f'(x) DNE and that means where it is undefined. Set the denominator equal to zero and solve

How to find the absolute extrema of f(x) on [a, b]

1. verify that the function is continuous on [a,b] 2. Find all the values of the critical points that are IN THE INTERVAL 3. evaluate the function at the critical points found in step 1 and the endpoints (plug in into function) 4. Identify the absolute extrema

What are critical numbers?

A critical number of a function f is a number c in the domain of f such that either f'(c)= 0 or f'(c) DNE

Fermat's Theorem

If f has a local maximum or minimum at c, and if f'(c) exists, then f'(c)=0

Conditions for definite integral

If f is CONTINUOUS on [a,b], or f only has a FINITE number of jump discontinuities, then f is integrable on [a,b] and the integral exists

Reimann Sum

If f is a function defined on [a,b] and Δis a partition of [a,b] where Δxi is the width of the i-th subinterval and ci is any point in the i-th subinterval then this equals ? ∑ ?(??)∆?? ?=1

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b]

If the derivative of a function is zero, what is happening to that function?

It is not changing

What does it mean when we say contained in an open interval when x is near c?

It means that we can find some interval (a,b) but not including the endpoints such that a<c<b Or, in other words, c will be contained somewhere inside the interval and will not be either of the endpoints.

Intuitive Definition of a Limit at Infinity

Let f be a function defined on some interval (a, infinity) Then the limit f(x) as x approaches infinity is L. This means that the values of f can be made arbitrarily close to L by requiring x to be sufficiently large

Are all functions integrable?

No

Can an endpoint be a local extrema? Why or why not?

No, end points can only be absolute extrema. The local extrema are located inside of an interval and so they require to be approached from both sides.

Does a horizontal tangent line gaurentee we will have a local maximum or minimum? Give an example

No, for example x^3 there is a horizontal tangent line but no maximum or minimum values

What are the extrema/extreme values of f?

The absolute maximum in minimum. Sometimes referred to as the global maximum or minimum

The function F is called an antideriative when...

The function F is called an antiderivative of f on an interval I of F'(x)- f(x) for all x I.

Mean Value Theorem

The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.

definite integral

The limit of the left-hand or right-hand sums with n subdivisions of [a, b] as n gets arbitrarily large.

True or false If f has a local maximum or minimum at c, then c is a critical number of f

True

Can there be an extreme value if f'(c) does not exist? Give an example

Yes, an example is the absolute value of x graph. There is an extreme value at 0; however, the derivative does not exist at that value.

Can functions change signs where they don't exist? Give example

Yes, for example 1/x which DNE at x=0 but it changes signs there.

What is the line y=L called?

a horizontal asymptote when lim f(x) as x approaches infinity or negative infinity equals L

If f'(x)=0 for every x on some interval I, then f(x) is __________ on the interval

constant

if f'(x) <0 for every x on some interval I, then f(x) is _____________ on the interval

decreasing

A critical point must be in the ________ of the function

domain

If the original function is increasing, the derivative is positive or negative?

f'(0)>0; the derivative is positive

If f'(x)<0 then

f(x) is decreasing on that interval

If f'(x)>0, then...

f(x) is increasing on that interval

If f takes on both positive and negative values, the Reimann sum....

gives us the net area. SUBTRACT the negative values from the positive values

If f'(x)>0 for every x on some interval I, then f(x) is __________ on the inerval

increasing

What does the extreme value theorem tell us?

that if f is continuous on a closed interval (including the endpoints), then we must have an absolute maximum and absolute minimum value

Definition of a Critical Point

we say that x=c is a critical point of the function f(x) if f(x) exists and if either of the following are true: 1. f'(c)=0 2. f'(c) DNE

At every local maximum or minimum, the derivative is equal to

zero