Unit One: Limits and Continuity- essential knowlegde

What concept does calculus use to understand and model dynamic change?

Calculus uses limits to understand and model dynamic change.

Why is the average rate of change undefined at a point where the change in the independent variable is zero?

Because an average rate of change divides the change in one variable by the change in another, the average rate of change is undefined at a point where the change in the independent variable would be zero.

How does the limit concept allow us to define the instantaneous rate of change?

The limit concept allows us to define the instantaneous rate of change in terms of average rates of change.

What is the definition of the limit of a function f(x) as x approaches c?

Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is lim f(x) = R.

In what multiple ways can a limit be expressed?

A limit can be expressed graphically, numerically, and analytically.

What does the concept of a limit include in terms of directional approach?

The concept of a limit includes one-sided limits.

How can graphical information about a function be used concerning limits?

Graphical information about a function can be used to estimate limits.

Why might graphical representations of functions miss important function behavior?

Because of issues of scale, graphical representations of functions may miss important function behavior.

Under what conditions might a limit not exist for some functions at particular values of x?

A limit might not exist if the function is unbounded, oscillating near this value, or if the limit from the left does not equal the limit from the right.

How can numerical information be used in relation to limits?

Numerical information can be used to estimate limits.

How can one-sided limits be determined?

One-sided limits can be determined analytically or graphically.

How can limits of sums, differences, products, quotients, and composite functions be found?

Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.

Why might it be necessary or helpful to rearrange functions before evaluating limits?

It may be necessary or helpful to rearrange functions using equivalent expressions into equivalent forms before evaluating limits.

What theorem can be used to find the limit of a function?

The limit of a function may be found by using the squeeze theorem.

What are the types of discontinuities in functions?

Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

What conditions must be met for a function f to be continuous at x = c?

A function f is continuous at x = c provided that f(c) exists, lim f(x) exists, and lim f(x) = f(c).

When is a function continuous on an interval?

A function is continuous on an interval if the function is continuous at each point in the interval.

Which types of functions are continuous on all points in their domains?

Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.

How can a discontinuity be removed if the limit of a function exists at that discontinuity?

If the limit of a function exists at a discontinuity in its graph, it is possible to remove the discontinuity by defining or redefining the function at that point, so it equals the value of the limit of the function as x approaches that point.

What must be true for a piecewise-defined function to be continuous at a boundary of its domain?

The value of the expression defining the function on one side of the boundary must equal the value of the expression defining the other side of the boundary, as well as the value of the function at the boundary.

How can the concept of a limit be extended to include infinite behavior?

The concept of a limit can be extended to include infinite limits.

How can asymptotic and unbounded behavior of functions be described?

Asymptotic and unbounded behavior of functions can be described and explained using limits.

How can the concept of a limit be extended regarding large values?

The concept of a limit can be extended to include limits at infinity.

What do limits at infinity describe about a function?

Limits at infinity describe end behavior.

How can the relative magnitudes of functions and their rates of change be compared?

Relative magnitudes of functions and their rates of change can be compared using limits.

What does the Intermediate Value Theorem guarantee for a continuous function on a closed interval?

If f is a continuous function on the closed interval [a, b] and d is a number between f(a) and f(b), there is at least one number c between a and b such that f(c) = d.