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, e^{x}. 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 xaxis. 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 xaxis. 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  
Function, f(x), y: y = x
 Slope Function, m, f'(x), dy/dx: m = 1
 Area Function, from a to b, is x^{2}/2, from a to b 
The Opposite Function, x  
Function, f(x), y: y = x
 Slope Function, m, f'(x), dy/dx: m =  1
 Area Function, from a to b, is  x^{2}/2, from a to b 
The Reciprocal Function, 1/x  
Function, f(x), y: y = 1/x
 Slope Function, m, f'(x), dy/dx: m = 1/(2x^{2})
 Area Function, from a to b, is lnx, from a to b 
A Constant Function, y = c, c = some constant  
Function, f(x), y: y = c, c = some constant
 Slope Function, m, f'(x), dy/dx: m = 0
 Area Function, from a to b, is c*ba, from a to b 
The "Twice" Function, y = 2x, as in two times the number  
Function, f(x), y: y = 2x
 Slope Function, m, f'(x), dy/dx: m = 2
 Area Function, from a to b, is x^{2}, from a to b 
The Squaring Function, y = x^{2}  
Function, f(x), y: y = x^{2}
 Slope Function, m, f'(x), dy/dx: m = 2x
 Area Function, from a to b, is x^{3}/3, from a to b 
The Square Root Function, y = x, is x^{1/2}  
Function, f(x), y: y =
x, is x^{1/2}
 Slope Function, m, f'(x), dy/dx: m = 1/(2
(x)), x > 0
 Area Function, from a to b, is 2* (x^{3})/3, from a to b 
The Absolute Value Function, x  
Function, f(x), y: y = x

 Area Function, from a to b, is x^{2}/2, from a to b 
An Exponential or Power Function, c^{x} where c > 0 and c 1, for example, 2^{3} means 2·2·2  
Function, f(x), y: y = c^{x}
 Slope Function, m, f'(x), dy/dx: m = c^{x}ln(c)
 Area Function, from a to b, is c^{x}/ln(c), from a to b 
The Exponential Function, exp(x) or e^{x}  
Function, f(x), y: y = exp(x) or e^{x}
 Slope Function, m, f'(x), dy/dx: m = e^{x}
 Area Function, from a to b, is e^{x}, from a to b 
On The function e^{x}, Its Derivative, & Its Integral 
A Lograthmic Functions, log_{c}(x)  
Function, f(x), y: y = log_{c}(x)
 Slope Function, m, f'(x), dy/dx: m = 1/(xln(c))
 Area Function, from a to b, is [xln(x)  x ]/ln(c), from a to b 
The Natual Log Function, ln(x) or log_{e}(x)  
Function, f(x), y: y = ln(x)
 Slope Function, m, f'(x), dy/dx: m = 1/x
 Area Function, from a to b, is [ xln(x)  x ], from a to b 
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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