Class Table library

Parent Functions, Their Slope Functions, and Area Functions



Intro

    This page was envisioned to give the reader exposure and play time with many functions, their slope functions (derivatives), and their area functions (integrals), prior to a discussion on the uniqueness of the exponential function, ex.

    In each colored computation box, enter the required info, press/click the gray button, see the resultant computation. Enter negatives as "-x" rather than "- x" on this page. Do NOT write input x values as fractions. Write fractions as good decimal approximations. CLICK ON THE GRAPH TO SEE AN ENLARGEMENT.

    Sometimes questions are written above or below a box. If possible, answer the question then, with the mouse, swipe the page between the two stars to see the answer to the question.

    To find the "area under a curve," from the point (a,0), draw a perpendicular to the point (a, f(a)) on the curve. One may need to draw it "up" to the curve or "down" to the curve. Repeat this procedure for b. Shade the area between x=a and x=b which is between the curve and the x-axis. This is the area needed. If instead the area between two curves (say f(x) and g(x)), from a to b, is needed, draw the perpendiculars x=a and x=b. Shade the area between the curves. The computation is covered on the derivative page listed below.

    An area may be negative. An area is negative if the curve falls below the x-axis. See the second example under "Arithmetic Tricks, Rules & Shortcuts In Words & Symbols" on page Derivatives and Integration.

    For other information, go to the pages listed below.

    For additional information on the function of this page and other functions see Function and Relation Library.   For additional information on trig functions see Trig Functions Library.

    Enjoy.



Functions Dislpayed on This Page

    The function names in words and the symbols for the function link to their graph, their slope and its graph, and their area function and its graph.

the identity function, x
the opposite function, -x
the reciprocal function, 1/x
a constant function, y = c, c = some constant
the "twice" function, y = 2x, as in two times the number
the squaring function, y = x2
the square root function, y = x, or x1/2
the absolute value function, |x|
an exponential or power function, cx where c > 0 and c 1,
      for example, 23 means 2·2·2
a lograthmic functions, logc(x), c > 0
the natual log function, ln(x) or loge(x)


The Identity Function, x
Function, f(x), y: y = x
x =
identityfx
Slope Function, m, f'(x), dy/dx: m = 1
x =
identityslope
Area Function, from a to b, is
x2/2, from a to b
a =      b =
identityINTEGRAL


The Opposite Function, -x
Function, f(x), y: y = -x
x =
oppositefx
Slope Function, m, f'(x), dy/dx: m = - 1
x =
oppositeslope
Area Function, from a to b, is
- x2/2, from a to b
a =      b =
oppositeINTEGRAL


1. What's the reciprocal of the reciprocal? * 1/(1/x) = x, the identify function*
The Reciprocal Function, 1/x
Function, f(x), y: y = 1/x
x =
reciprocalfx
Slope Function, m, f'(x), dy/dx: m = -1/(2x2)
x =
reciprocalslope
Area Function, from a to b, is
ln|x|, from a to b
a =      b =
reciprocalINTEGRAL


1. In the case below, what's the value of c, the constant?*c is 2*
A Constant Function, y = c, c = some constant
Function, f(x), y: y = c, c = some constant
x =      c=
constantfx
Slope Function, m, f'(x), dy/dx: m = 0
x =     c=
constantslope
Area Function, from a to b, is
c*|b-a|, from a to b
a =      b =     c =
constantINTEGRAL


The Doubling Function, y = 2x, as in two times the number
x =
doublingfx
Slope Function, m, f'(x), dy/dx: m = 2
x =
doublingslope
Area Function, from a to b, is
x2, from a to b
a =      b =
doublingINTEGRAL


The Squaring Function, y = x2
Function, f(x), y: y = x2
x =
squaringfx
Slope Function, m, f'(x), dy/dx: m = 2x
x =
squaringslope
Area Function, from a to b, is
x3/3, from a to b
a =      b =
squaringINTEGRAL


The Square Root Function, y = x, is x1/2
Function, f(x), y: y = x, is x1/2
x =
squarerootfx
Slope Function, m, f'(x), dy/dx: m = 1/(2 (x)), x > 0
x =
squarerootslope
Area Function, from a to b, is
2* (x3)/3, from a to b
a =      b =
squarerootINTEGRAL


The Absolute Value Function, |x|
Function, f(x), y: y = |x|
x =
absolutevaluefx
absolutevaluefxsymbols
x =

If x is 0, the slope is 0.
absoluteslope
Area Function, from a to b, is
x2/2, from a to b
a =      b =
absoluteINTEGRAL


1. In this case, what is the value of c, the base, used in the graphs?* c is 2*
An Exponential or Power Function, cx where c > 0 and c 1, for example, 23 means 2·2·2
Function, f(x), y: y = cx, c 1
x =     c =
2tothexfx
Slope Function, m, f'(x), dy/dx: m = cxln(c)
x =     c = c>0
2tothexslope
Area Function, from a to b, is
cx/ln(c), from a to b
x =     c =  , c>0     a =     b =
      
2tothexINTEGRAL


1. How does the function in this table compair to the function in the last table?
      *Above the base is c. Below the base is e.*
2. What is the value of ln(e)?*It is 1.*
The Exponential Function, exp(x) or ex
Function, f(x), y: y = exp(x) or ex
x =
exponentialfx
Slope Function, m, f'(x), dy/dx: m = ex
x =
exponentialslope
Area Function, from a to b, is
ex, from a to b
a =     b =
etothexINTEGRAL
3. How does the slope function in this table compair to the slope function in the last table?
      *Above m = cxln(c). Below m = exln(e) which is ex.*
4. How does the area function in this table compair to the area function in the last table?
      *Above area = cx/ln(c). Below area = ex/ln(e) which is ex.*
5. What may be said about the function, the slope function, and the area function in the above table?
      *They are each ex.*



1. What is a log, logarithm? *A log is an exponent.*
2. The symbols "log2(8)" is read "the exponent to which 2 must be raised in order to get 8."
     What is the value of log2(8)?
*3, because 23 = 8*
3. What is c, the base of this log function? *c = 2*
A Lograthmic Functions, logc(x)
Function, f(x), y: y = logc(x)
x =       c =
logbase2(x)fx
Slope Function, m, f'(x), dy/dx: m = 1/(xln(c))
x =       c =
logbase2(x)slope
Area Function, from a to b, is
 [xln(x) - x ]/ln(c), from a to b
c =      a =      b =
logbase2(x)INTEGRAL





1. What is the value of log10(10)?*It is 1.*
2. What is the value of ln(e)?*It is 1.*
The Natual Log Function, ln(x) or loge(x)
Function, f(x), y: y = ln(x)
x =
naturallogfx
Slope Function, m, f'(x), dy/dx: m = 1/x
x =
naturallogslope
Area Function, from a to b, is
[ xln(x) - x ], from a to b
a =      b =
naturallogINTEGRAL

In Closing ...

    Hope you had some fun.

    For more about "Exponential or Power Functions," "A Bit about e, the base of the Natural Logs," or "The Exponential Function," visit this page in the Function and Relation Library.



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