Mathematica Instructions

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

Introductory Instructions:

  1. Download Mathematica
  2. Basic Functions (note the capitalization of first letter)
  3. To execute mathematica code


  1. Evaluate f(x) at x = a
  2. Evaluate f(x) at x = a
  3. Evaluate
  4. Define piecewise-defined function
  5. Plot a function
  6. Plot a function
  7. Plot the real part of a radical function
  8. Plot multiple functions on same graph
  9. Implicitly plot an equation
  10. Factor/Simplify an expression
  11. Simplify an expression
  12. Expand an expression
  13. Combine rational expressions
  14. Find real roots of polynomial equations


  1. Expand a trigonometric expression
  2. Find real roots of trignometric equation (if NSolve does not work)

Section 2.2:

  1. Evaluate bidirectional limits
  2. Evaluate limits from left
  3. Evaluate limits from right

Section 2.5:

  1. Evaluate limits at infinity

Section 2.6:

  1. Find the derivative at a point x = a

Section 2.7:

  1. Find the derivative function

Section 2.8:

  1. Multiple derivatives
  2. Graph derivative functions
  3. Find location(s) of local maximum(s) and local minimum(s)
  4. Find location(s) of inflection point(s)

Section 1.7:

  1. Find the parametric equations

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

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

Section 5.2

  1. Evaluating sums (e.g., sum of i from 1 to 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 xi 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

Section 5.4:

  1. Fundamental Theorem of Calculus, Part 1 (FTC1)

Still need to code: