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Differential Equation Model

"If a function is growing or shrinking exponentially, it can be modeled using a differential equation. The equation itself is dy/dx=ky, which leads to the solution of y=ce^(kx). In the differential eq ...

Derivatives of Logarithmic Functions

"The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarith ...

Derivatives of Exponential Functions

"The derivative of an exponential function can be derived using the definition of the derivative. Derivatives of exponential functions involve the natural logarithm function, which itself is an impor ...

Derivative Notation

"The two commonly used ways of writing the derivative are Newton's notation and Liebniz's notation. Newton's notation involves a prime after the function to be derived, while Liebniz's notation utiliz ...

Derivative Definition

"The definition of the derivative is the slope of a line that lies tangent to the curve at the specific point. The limit of the instantaneous rate of change of the function as the time between measure ...

Definite Integrals as Net Change

"Definite integrals can be used to find the net change in a function between two times. The fundamental theorem of Calculus can be restated so that the definite integral of the function's derivative i ...

Definite Integrals and Area

"Definite integrals can be used to find the area under, over, or between curves. If a function is strictly positive, the area between it and the x axis is simply the definite integral. If it is simply ...

Quotient Rule

"The quotient rule is useful when trying to find the derivative of a function that is divided by another function. As long as both functions have derivatives, the quotient rule tells us that the final ...

Curve Sketching

"Sketching a curve from knowledge of the signs of the first and second derivatives is a useful way to find the approximate shape of a function's graph. When curve sketching making a sign chart of the ...

Critical Points

"Critical points are places where the derivative of a function is either zero or undefined. These critical points are places on the graph where the slope of the function is zero. All relative maxima a ...

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