MA 16100 Exam Instructions

The derivative of a function at a point represents the instantaneous rate of change of the function at that point. It's the slope of the tangent line to the function's graph at that point.

The tangent line to a curve at a point is a straight line that touches the curve at that point and has the same slope as the curve at that point.

The derivative of a function f(x) is denoted as f'(x) and is defined as the limit of the difference quotient as the change in x approaches zero.

A function is increasing on an interval if its derivative is positive on that interval.

A function is decreasing on an interval if its derivative is negative on that interval.

A critical point of a function is a point where the derivative is zero or undefined.

A function is concave up on an interval if its second derivative is positive on that interval.

A function is concave down on an interval if its second derivative is negative on that interval.

An inflection point of a function is a point where the concavity changes, from concave up to concave down, or vice versa.

The definite integral of a function f(x) from a to b represents the area under the curve of f(x) between x = a and x = b.

The Fundamental Theorem of Calculus connects differentiation and integration. It states that the derivative of the definite integral of a function is the original function.

The indefinite integral of a function f(x) is a family of functions whose derivative is f(x). It's denoted as ∫f(x)dx.

A limit is the value that a function approaches as its input approaches a certain value.

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

A vertical asymptote is a vertical line that the graph of a function approaches as x approaches a certain value where the function becomes unbounded.

The derivative of a composite function f(g(x)) can be found using the chain rule, which states that the derivative is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

The absolute maximum of a function on an interval is the largest value the function takes on that interval.

The absolute minimum of a function on an interval is the smallest value the function takes on that interval.

The Mean Value Theorem states that for a differentiable function on a closed interval, there exists a point within that interval where the slope of the tangent line is equal to the average slope of the function over that interval.

The derivative of a function represents the instantaneous rate of change of the function at a point. It is the slope of the tangent line to the graph of the function at that point.

An exponential function is a function where the independent variable appears in the exponent. It has the general form f(x) = a^x, where a is a constant.

A logarithmic function is the inverse of an exponential function. It is used to find the exponent to which a base must be raised to obtain a given value. It has the general form f(x) = log_a(x), where a is the base.

The natural logarithm is a special case of the logarithmic function where the base is e (Euler's number). It is denoted as ln(x).

The inverse of a function reverses the input and output. If f(a) = b, then f^-1(b) = a.

A function is one-to-one if each input value corresponds to a unique output value. This ensures that the function has an inverse.

Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.

Related rates problems involve finding the rate of change of one quantity with respect to another quantity when both quantities are changing.

A sphere is a three-dimensional geometric shape that is perfectly round. It is defined as the set of all points that are equidistant from a given point called the center.