Wolfram Alpha Examples M071
Wolfram alpha is available at http://www.wolframalpha.com/. There is also an Iphone app at http://products.wolframalpha.com/iphone/ and an Android app at http://products.wolframalpha.com/android/ . See http://www.wolframalpha.com/examples/Math.html for a list of examples for mathematical problems that Wolfram alpha can solve. Also see http://www.wolframalpha.com/examples/ for example of other problems that Wolfram alpha can solve.
A tip for Wolfram Alpha input: to help Wolfram Alpha understand your problem it is better to include spaces between consecutive variables. For example, the formula for the volume of a box with side x, y and z, is better input as x y z rather than xyz. Extra spaces are a good habit for all Wolfram Alpha input.

Example: find a limit.
(1) Original problem: _{}
(2) Wolfram alpha input: lim ( x^2  4)/(x 2) as x>2
(3) Wolfram alpha output:
To get the output and paste it into word: (a) right click on the Wolfram output in your browser, (b) select copy image, (c) move to MS word and paste (ctrl V).

Example: calculate a derivative.
(1) Original problem: _{}
(2) Wolfram alpha input: d/dx sqrt(x^2 + x1)/( x^21)
(3) Wolfram alpha output:

Example: find the slope of a tangent line:
(1) Original problem: find the slope of the tangent line to _{} at x=2
(2) Wolfram alpha input: d/dx sqrt(x^2 + x1)/( x^21) at x=2
(3) Wolfram alpha output:

Example: find a second derivative.
(1) Original problem: _{}
(2) Wolfram alpha input: d^2/dx^2 sqrt(x^2 + x1)/(x^21)
(3) Wolfram alpha output:

Example: implicit differentiation.
(1) Original problem: Find _{} if _{}
(2) Wolfram alpha input: find dy/dx if x y^2+4 x y = 10
(3) Wolfram alpha output:

Example: critical numbers.
(1) Original problem: Find the critical numbers of _{}
(2) Wolfram alpha input: d/dx 2 x^39 x^2
(3) Wolfram alpha output (scroll down to Roots): and
It was necessary to use copy image twice on the Wofram alpha page, once for each root, and to paste each root to this word document.

Example: draw a plot of a function.
(1) Original problem: draw a plot of the function _{}
(2) Wolfram alpha input: 2 x^39 x^2
(3) Wolfram alpha output (scroll down to Plots):

Example: indefinite integral
(1) Original problem: find the indefinite integral _{}
(2) Wolfram alpha input: integral exp(x) / ( 1 + exp(x)) dx
(3) Wolfram alpha output:

Example: definite integral
(1) Original problem: find the indefinite integral _{}
(2) Wolfram alpha input: integral exp(x) / ( 1 + exp(x)) dx, x= 0 .. 2
(3) Wolfram alpha output:

Example: plot a surface and a contour plot

Original problem: draw a plot of the surface _{} in 3 dimensions and draw a contour plot in 2 dimensions

Wolfram alpha input: 2 x^3+x y^2+5 x^2+y^2

Wolfram alpha output (scroll down to and copy 3 D plot and Contour plot)

Example: partial derivatives
(1) Original problem: evaluate _{} of _{} for _{}
(2) Wolfram alpha input: d/dx d/dy x exp(x+y)
(3) Wolfram alpha output:

Example: solution to a Lagrange multiplier problem
(1) Original problem: use Lagrange multipliers to maximize V = x y z subject to the constraint_{}. See the example on page 958 of Larson and Hodgkins. Instead of the Greek letter λ we use the English w.
We present three different ways to solve the problem. More background discussion of these three solutions is in the document titled “Lagrange Multiplier Examples” at http://www.math.sjsu.edu/~foster/m071/m071fall12.html .
First Solution: Solve the Lagrange equations (the equations obtained by setting all partial derivatives of F(x,y,z,w) = x y z – w ( 6 x +4 y+3 z – 24) to zero.)
Wolfram Alpha Input: solve y z  6w = 0, x z4w = 0, x y – 3 w = 0, 6 x – 4 y – 3 z + 24 = 0
Wolfram Alpha Output:
The correct answer to the maximization problem is the last solution since this solution, with x = 4/3, y = 2 and z = 8/3, has V = 64/9 which is larger than V values (= 0) for the other solutions.
OR
Second Solution: find a stationary point of the Lagrange function F. A stationary point is a point where all the partial derivatives of a function are zero.
(2) Wolfram alpha input (note the space between the w and the left parenthesis is required): stationary points of x y z – w ( 6 x +4 y+3 z – 24)
(3) Wolfram alpha output:
Substituting each of the four possible (x,y,z) values into V, we see that V=(4/3)(2)(8/3)=64/9 is the maximum value of V.
OR
Third Solution: solve a constrained optimization problem. In the problem we want to maximize xyz subject to the constraint that Implicit in the problem are additional constraints that since V is the volume of a box and the box must have sides with nonnegative length.
(2) Wolfram Alpha input: maximize x y z subject to 6 x + 4 y + 3 z  24 = 0, x >= 0, y >= 0, z >= 0
(3) Wolfram Alpha output:
Here the is Wolfram Alpha’s symbol for “and.” 