Mathematica Instructions

Here are instructions on how to use Mathematica for Calculus I:

Introductory Instructions:

1. Download Mathematica
   • https://www.gpc.edu/servicedesk/mathematica.htm
2. Basic Functions (note the capitalization of first letter)
   • For e^x use: Exp[x]
   • For π use: Pi
   • For sin(x) use: Sin[x] – all other trigonometric functions are similar
   • For sin^–1(x) use: ArcSin[x] – all other trigonometric functions are similar
   • For | x | use: Abs[x]
   • For √x use: Sqrt[x]
   • For ∞ use: Infinity
   • For ln(x) use: Log[x]
   • For log_b(x) use: Log[b, x]
   • For ≤ use: <=
   • For ≥ use: >=
   • For sinh(x) use: Sinh[x] – all other hyperbolic functions are similar
   • For sinh^–1(x) use: ArcSinh[x] – all other inverse hyperbolic functions are similar
3. To execute mathematica code
   • Hold SHIFT key and press ENTER

Algebra:

1. Evaluate f(x) at x = a
   • Define function using: f[x_] := expression, then use: f[a]
     note: expression is any algebraic expression
     note: After evaluating, you should clear the function using: Clear[f]
2. Evaluate f(x) at x = a
   • expression /. x -> a
     note: expression is any algebraic expression
     note: You can quickly evaluate the expression at several values of x using: expression /. x -> {a1, a2, a3} [where a1, a2, a3, etc. are different x-values]
3. Evaluate
   • N[expression, d]
     note: expression is any expression to be evaluated
     note: d is the number of digits needed in answer
4. Define piecewise-defined function
   • Define function using: f[x_] := Piecewise[{{function1, condition1},{function2, condition2}}, {x}]
5. Plot a function
   • Plot[expression, {x, xmin, xmax}]
     note: To change the y-values shown on the graph, use the additional argument: PlotRange -> {ymin, ymax}
     note: To center the graph at the origin, use the additional argument: AxesOrigin -> {0, 0}
6. Plot a function
   • Define a function (see above), then use: Plot[f[x], {x, xmin, xmax}]
7. Plot the real part of a radical function
   • Plot[Sign[x]*Abs[x]^(1/n), {x, xmin, xmax}]
     note: 1/n is the n^th root function
8. Plot multiple functions on same graph
   • Plot[{expression1, expression2, expression3}, {x, xmin, xmax}]
     note: expression1, expression2, expression3, etc. are different functions
9. Implicitly plot an equation
   • ContourPlot[expression1 == expression2, {x, xmin, xmax}, {y, ymin, ymax}, AxesOrigin -> {0, 0}]
     note: the last two arguments may not be needed
10. Factor/Simplify an expression
    • Factor[expression]
11. Simplify an expression
    • Simplify[expression, var]
      note: var is the independent variable
    • FullSimplify[expression, var]
      note: var is the independent variable
12. Expand an expression
    • Expand[expression]
13. Combine rational expressions
    • Together[expression1 + expression2]
      note: You may have more than two rational expressions and can add or subtract the expressions
14. Find real roots of polynomial equations
    • Solve[polynomial == 0, x, Reals]
      note: For approximate solutions, use NSolve[polynomial == 0, x, Reals]

Precalculus:

1. Expand a trigonometric expression
   • TrigExpand[expression]
2. Find real roots of trignometric equation (if NSolve does not work)
   • Plot function, then use: FindRoot[expression == 0, {x, xval}]
     note: xval is an x-value near the x-intercept of the function

Section 2.2:

1. Evaluate bidirectional limits
   • Limit[expression, var -> a]
2. Evaluate limits from left
   • Limit[expression, var -> a, Direction -> 1]
3. Evaluate limits from right
   • Limit[expression, var -> a, Direction -> -1]

Section 2.5:

1. Evaluate limits at infinity
   • Limit[expression, var -> Infinity]

Section 2.6:

1. Find the derivative at a point x = a
   • Define a function (see above), then use: Limit[(f[a + h] - f[a])/h, h -> 0]; OR
   • Define a function (see above), then use: f'[a]

Section 2.7:

1. Find the derivative function
   • Define a function (see above), then use: Limit[(f[x + h] - f[x])/h, h -> 0]; OR
   • D[function, var]
     note: var is the independent variable; OR
   • Define a function (see above), then use: f'[x]

Section 2.8:

1. Multiple derivatives
   • D[function, {var, n}]
     note: var is the independent variable and n is the n^th derivative; OR
   • Define a function (see above), then use: f''[x]
     note: use as many ' as needed
2. Graph derivative functions
   • Define a function (see above), then use: Plot[f'[x], {x, xmin, xmax}]
     note: you can graph any multiple derivative by using as many ' as needed
3. Find location(s) of local maximum(s) and local minimum(s)
   • Define a function (see above), then use: NSolve[f'[x] == 0, x, Reals]
4. Find location(s) of inflection point(s)
   • Define a function (see above), then use: NSolve[f''[x] == 0, x, Reals]

Section 1.7:

1. Find the parametric equations
   • ParametricPlot[{x-function, y-function}, {t, tmin, tmax}]

Section 3.5:

1. Implicit differentiation
   1. Store the equation into the variable equation1: equation1 = expression1 == expression2, then SHIFT-ENTER
   2. Convert all y-variables into a function of x, i.e., y[x], and store into new variable: equation2 = equation1 /. y -> y[x], then SHIFT-ENTER
   3. Differentiate each side of the equation and store into new variable: equation3 = D[equation2[[1]], x] == D[equation2[[2]], x], then SHIFT-ENTER
   4. Solve the last equation for y'[x]: Solve[equation3, y'[x]], then SHIFT-ENTER

Section 4.5:

1. l'Hôpital's Rule
   • Define numerator function as f[x_] and denominator function as g[x_] (see above), then use: Limit[f'[x]/g'[x], x -> a]

Section 4.7:

1. Newton's Method
   1. Define function as f[x_] (see above)
   2. copy and paste the entire algorithm below, then SHIFT-ENTER:
      
      

      recursiveNewton[f_, x_?NumberQ, previousX_: 0, tolerance_: (N[10^-20])] := 
       If[Abs[x - previousX] > tolerance, Print[N[x, 20]];
        recursiveNewton[f, N[x - f[x]/f'[x], 30], x, tolerance], x  (*return result*)]

      
      
   3. then use: recursiveNewton[f, guess]
      note: where guess is an x-value near the root of the function

Section 4.8:

1. Antiderivatives/Indefinite Integrals
   • Define numerator function as f[x_] (see above), then use: Integrate[f[x], x]; OR
   • Integrate[expression, x]

Section 5.2

1. Evaluating sums (e.g., sum of i from 1 to n)
   • Sum[i, {i, 1, n}]
     
2. Evaluating Riemann sums
   1. Store integrand into f(x) using: f[x_]:=function
   2. Store Δx into variable dx using: dx=(b-a)/n – where a is lower limit of integration and b is upper limit of integration
   3. Find x_i and store into variable xi using: xi=expression
   4. Evaluate the sum and store into sum variable: sum = Sum[f[xi]*dx,{i,1,n}]
   5. Evaluate the limit of the sum as n → ∞: Limit[sum, n -> Infinity]

Section 5.2 & 5.3:

1. Evaluating definite integrals
   • Define function as f[x_] (see above), then use: Integrate[f[x], {x, a, b}]
     note: where a is the lower limit of integration and b is the upper limit of integration; OR
   • Integrate[expression, {x, a, b}]
     note: NIntegrate can be used to find an approximation of the definite integral

Section 5.4:

1. Fundamental Theorem of Calculus, Part 1 (FTC1)
   • Define function as f[t_] as function (see above), then use: D[Integrate[f[t], {t, 1, g(x)}], x] – where g(x) is some function of x

Still need to code:

• Graphing Riemann sums (Section 5.1)