difference quotient

Examples of difference quotient in the following topics:

• The Derivative and Tangent Line Problem

  • The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
  • As the point $q$ approaches $p$, which corresponds to making $h$ smaller and smaller, the difference quotient should approach a certain limiting value $k$, which is the slope of the tangent line at the point $p$.
  • Then there is a unique value of $k$ such that, as $h$approaches $0$, the difference quotient gets closer and closer to $k$, and the distance between them becomes negligible compared with the size of $h$, if $h$ is small enough.
  • This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function $f$.

• Differentiation Rules

  • In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.

• Differentials

  • holds, where the derivative is represented in the Leibniz notation $\frac{dy}{dx}$, and this is consistent regarding the derivative as the quotient of the differentials.

• Calculating Limits Using the Limit Laws

  • The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily.

• Derivatives of Logarithmic Functions

  • We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such as products with many terms, quotients of composed functions, or functions with variable or function exponents.

• Four Ways to Represent a Function

  • Modern calculus texts emphasize that a function can be expressed in four different ways.
  • These are not four different types of functions; they are four different views of the same function.
  • One of the most important skills in algebra and calculus is being able to convert a function between these different forms, and this theme will recur in different forms throughout the text.

• Limits and Continuity

  • For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola.
  • Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

• Sequences

  • Unlike a set, order matters in a sequence, and exactly the same elements can appear multiple times at different positions in the sequence.
  • Examples: $(M, A, R, Y)$ is a different sequence from $(A, R, M, Y)$.
  • Also, the sequence $(1, 1, 2, 3, 5, 8)$, which contains the number $1$ at two different positions, is a valid sequence.

• Iterated Integrals

  • It is key to note that this is different, in principle, from the multiple integral $\iint f(x,y)\,dx\,dy$ .
  • While the antiderivatives of single variable functions differ at most by a constant, the antiderivatives of multivariable functions differ by unknown single-variable terms, which could have a drastic effect on the behavior of the function.

• Indefinite Integrals and the Net Change Theorem

  • Basically, the theorem states that the integral of or $F'$ from $a$ to $b$ is the area between $a$ and $b$, or the difference in area from the position of $f(a)$ to the position of $f(b)$.
  • Apply the basic properties of indefinite integrals, including the constant, sum, and difference rules