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Boundless Calculus
Derivatives and Integrals
Applications of Differentiation
Calculus Textbooks Boundless Calculus Derivatives and Integrals Applications of Differentiation
Calculus Textbooks Boundless Calculus Derivatives and Integrals
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 8
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Derivatives and the Shape of the Graph

The shape of a graph may be found by taking derivatives to tell you the slope and concavity.

Learning Objective

  • Sketch the shape of a graph by using differentiation to find the slope and concavity


Key Points

    • The derivative of a function is the the function that defines the slope of the graph at each point.
    • The second derivative of the graph tells you the concavity of the graph at a point.
    • Inflection points are where the second derivative is 0 and are points where the concavity changes.

Terms

  • concave

    curved like the inner surface of a sphere or bowl

  • convex

    curved or bowed outward like the outside of a bowl or sphere or circle


Full Text

Differentiation is a method to compute the rate at which a dependent output $y$ changes with respect to the change in the independent input $x$. This rate of change is called the derivative of $y$ with respect to $x$. In more precise language, the dependence of $y$ upon $x$ means that $y$ is a function of $x$. This functional relationship is often denoted $y=f(x)$, where $f$ denotes the function.

If $x$ and y are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.

The simplest case is when $y$ is a linear function of $x$, meaning that the graph of $y$ divided by $x$ is a straight line. In this case, $y=f(x) = m \cdot x+b$, for real numbers $m$ and $b$, and the slope m is given by $\frac{\Delta y}{\Delta x}$, where the symbol $\Delta$ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in. "

This formula is true because:

$y + \Delta y $

$= f(x+ \Delta x) $

$= m (x + \Delta x) + b $

$= m (x + \Delta x) + b $

$= y + m \Delta x$

It follows that:

$\Delta y = m \Delta x$

This gives an exact value for the slope of a straight line. If the function $f$ is not linear (i.e. its graph is not a straight line), however, then the change in $y$ divided by the change in $x$ varies: differentiation is a method to find an exact value for this rate of change at any given value of $x$.

Inflection Point

A point where the second derivative of a function changes sign is called an inflection point. At an inflection point, the second derivative may be zero, as in the case of the inflection point $x=0$ of the function $y=x^3$, or it may fail to exist, as in the case of the inflection point $x=0$ of the function $y=x^{\frac{1}{3}}$. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.

Derivative

At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where a green line appears, negative where a red line appears, and zero where a black line appears.

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