# Jaddybill2

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#### Limits(rate of change & instanteous velocity)

If f is a function of x, then the instantaneous rate of change at x=a is the limit of the average rate of change over a short interval, as we make that interval smaller and smaller. ... This is the slope of the line tangent to y=f(x) at the point (a,f(a))

- Book
- Class notes
- • 9 pages •

If f is a function of x, then the instantaneous rate of change at x=a is the limit of the average rate of change over a short interval, as we make that interval smaller and smaller. ... This is the slope of the line tangent to y=f(x) at the point (a,f(a))

#### Limits of functions

A limit tells us the value that a function approaches as that function's inputs get closer and closer to some number. The idea of a limit is the basis of all calculus.

- Book
- Class notes
- • 8 pages •

A limit tells us the value that a function approaches as that function's inputs get closer and closer to some number. The idea of a limit is the basis of all calculus.

#### Finding limits by evaluation

Evaluating Limits. Just Put The Value In. The first thing to try is just putting the value of the limit in, and see if it works (in other words substitution). Factors. We can try factoring. Conjugate. Infinite Limits and Rational Functions. L'Hôpital's Rule. Formal Method.

- Book
- Class notes
- • 9 pages •

Evaluating Limits. Just Put The Value In. The first thing to try is just putting the value of the limit in, and see if it works (in other words substitution). Factors. We can try factoring. Conjugate. Infinite Limits and Rational Functions. L'Hôpital's Rule. Formal Method.

#### Limit laws

The limit of a constant times a function is equal to the constant times the limit of the function. The limit of a product is equal to the product of the limits. The limit of a quotient is equal to the quotient of the limits. The limit of a constant function is equal to the constant.

- Book
- Class notes
- • 11 pages •

The limit of a constant times a function is equal to the constant times the limit of the function. The limit of a product is equal to the product of the limits. The limit of a quotient is equal to the quotient of the limits. The limit of a constant function is equal to the constant.

#### Limits by algebric manipulation and one sided limits

A one-sided limit is the value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1.

- Book
- Class notes
- • 14 pages •

A one-sided limit is the value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1.

#### Continuity and Discontinuity

A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function's value. ... Jump discontinuity is when the two-sided limit doesn't exist because the one-sided limits aren't equal.

- Book
- Class notes
- • 8 pages •

A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function's value. ... Jump discontinuity is when the two-sided limit doesn't exist because the one-sided limits aren't equal.

#### Infinite limits

In general, a fractional function will have an infinite limit if the limit of the denominator is zero and the limit of the numerator is not zero. ... As x approaches 2 from the left, the numerator approaches 5, and the denominator approaches 0 through negative values; hence, the function decreases without bound and .

- Book
- Class notes
- • 12 pages •

In general, a fractional function will have an infinite limit if the limit of the denominator is zero and the limit of the numerator is not zero. ... As x approaches 2 from the left, the numerator approaches 5, and the denominator approaches 0 through negative values; hence, the function decreases without bound and .

#### Limits at infinity and Tangent lines

Limits at infinity are used to describe the behavior of functions as the independent variable increases or decreases without bound. If a function approaches a numerical value L in either of these situations, write. and f( x) is said to have a horizontal asymptote at y = L.) Find the first derivative of f(x). 2) Plug x value of the indicated point into f '(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point...

- Book
- Class notes
- • 11 pages •

Limits at infinity are used to describe the behavior of functions as the independent variable increases or decreases without bound. If a function approaches a numerical value L in either of these situations, write. and f( x) is said to have a horizontal asymptote at y = L.) Find the first derivative of f(x). 2) Plug x value of the indicated point into f '(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point...

#### Differentiation

The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.

- Book
- Class notes
- • 10 pages •

The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.

#### Derivative as a function

Learn rules of differentiation and derivatives as a function

- Book
- Summary
- • 13 pages •

Learn rules of differentiation and derivatives as a function