Differential Calculus short conceptual (gemini)

Which term refers to a relation that assigns exactly one element of set Y to each element of set X?

Function

Which of the following best describes the domain of a function?

The set of all values for which the function is defined

What is the range of a function?

The set of all possible output values of the function

What is the value that a function approaches as the input approaches a certain value, regardless of the function's actual value at that point?

Limit

What is required for the limit of a function to exist at a point?

The left-hand and right-hand limits must be equal

The expression $1^\infty$ that appears in limits is an example of what kind of form?

Indeterminate form

List the seven Indeterminate Forms.

$0/0$, $\infty/\infty$, $0 \times \infty$, $0^0$, $\infty^0$, $1^\infty$, $\infty - \infty$

Which of the following conditions must be satisfied to determine if a function is continuous at a number $a$?

All of these are correct: $f(a)$ is defined; $\lim{x\to a} f(x)$ exists; and $\lim{x\to a} f(x) = f(a)$

What type of discontinuity occurs when the limit exists but does not equal the function value?

Removable discontinuity

If a function has different left-hand and right-hand limits at a point, it has:

Jump discontinuity

What does the derivative of a function at a point represent?

The slope of the tangent line at that point

If the derivative of a function is positive over an interval, what can be concluded?

The function is increasing

The point where the derivative changes from positive to negative is a:

Local maximum

Which of the following statements correctly describe a critical point of a function?

A point where the first derivative is zero AND a point where the first derivative is undefined, but the function is defined

What does the second derivative of a function describe?

The concavity of the function

If $f''(x) > 0$ over an interval, what does this indicate about the graph?

Concave up

If $f''(x) < 0$ over an interval, what does this indicate about the graph?

Concave down

What condition must be met at a point for it to be considered a possible inflection point?

$f''(x) = 0$ and changes sign

How do you find the slope ($dy/dx$) of a parametric curve?

Divide $dy/dt$ by $dx/dt$

What does a partial derivative represent?

The rate of change of a function with respect to one variable while holding others constant

What does Clairaut’s Theorem state?

$\partial^2f/\partial x\partial y = \partial^2f/\partial y\partial x$ (Mixed second partial derivatives are equal)