Mathematica Instructions

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

Introductory Instructions:

1. Download Mathematica
   • https://www.gpc.edu/servicedesk/mathematica.htm
2. For basic mathematica usage instructions; using mathematica for algebra; using mathematica for precalculus; using mathematica for calculus I:
   • http://calc.jjw3.com/math2431/mathematica_instructions.htm

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]
3. Graphing Riemann sums
   • See Project 1:
     

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
2. Shading area under the curve
   • Define function as f[x_] (see above), then use: Plot[f[x], {x, a, b}, Filling -> Axis, FillingStyle -> Hue[d]]
     note: where [a, b] is a subset of the domain; and 0 ≤ d ≤ 1.

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

Section 5.7 & Appendix G:

1. Partial Fractions
   • Define rational function as f[x_] as function (see above), then use: Apart[[f[x]]]
2. Solving Systems of Equations [needed to algebraically find partial fractions]
   • To solve the system, expr1 = a, expr2 = b, expr3 = c, etc, use: Solve[{expr1, expr2, expr3} == {a, b, c}]
     NOTE: You must have the same number of equations as variables; and the order of the variables in each expression is not important.

Section 5.9:

1. Approximate Integration
   • See Project 1:

Section 5.10:

1. Improper Integrals
   1. Infinite Intervals:
      • Formal Method: Use Limit function
        • Limit[Integrate[f[x], {x, a, b}], a -> -Infinity] where b is the value for the upper limit of integration
        • Limit[Integrate[f[x], {x, a, b}], b -> Infinity] where a is the value for the lower limit of integration
        • NOTE: If a = –∞ and b = ∞, then you need to split the integral into two different integrals using x = c and use both of the above improper integrals
          NOTE: If Mathematica returns a number, then the improper integral converged to that number; if Mathematica returns either –∞ or ∞, then the improper integral diverged.
      • Informal Method: Use Integrate function directly
        • Integrate[f[x], {x, a, b}] where a = –∞, b = ∞, or both
        • NOTE: If Mathematica returns a number, then the improper integral converged to that number; if Mathematica returns an error message including the phrase "does not converge on", then the improper integral diverged.
   2. Discontinuous Integrands:
      • Formal Method: Use Limit function
        • Limit[Integrate[f[x], {x, a, b}], a -> c, Direction -> -1] where f(x) is not continuous at x = c and b is the value for the upper limit of integration
        • Limit[Integrate[f[x], {x, a, b}], b -> c, Direction -> 1] where f(x) is not continuous at x = c and a is the value for the lower limit of integration
        • NOTE: If f(x) is not continuous at x = d for some d in the interval (a, b), then you need to split the integral into two different integrals using x = d and use both of the above improper integrals
          NOTE: If Mathematica returns a number, then the improper integral converged to that number; if Mathematica returns either –∞ or ∞, then the improper integral diverged.
      • Informal Method: Use Integrate function directly
        • Integrate[f[x], {x, a, b}] where f(x) is not continuous at x = a, x = b, or x = d in the interval (a, b)
        • NOTE: If Mathematica returns a number, then the improper integral converged to that number; if Mathematica returns an error message including the phrase "does not converge on", then the improper integral diverged.

Section 6.1:

1. Shading area between two curves
   • Define function as f[x_] and g[x_] (see above), then use: Plot[{f[x], g[x]}, {x, xmin, xmax}, Filling -> {1 -> {{2}, Hue[d]}}]
     note: where [a, b] is a subset of the domain; 0 ≤ d ≤ 1; and only change the value for d in the Filling option (i.e., do not change the 1 and 2 in the Filling option)
     
2. Plot two functions of y
   • Define function as f[y_] and g[y_] (see above), then use: ContourPlot[{x == f[y], x == g[y]}, {x, xmin, xmax}, {y, ymin, ymax}, Frame -> False, Axes -> True]
3. Find the intersection(s) of two functions of y
   • Define function as f[y_] and g[y_] (see above), then use: Solve[f[y] == g[y], y]
   • note: To find the x-coordinates, evaluate f(y) at each value calculated above

Appendix H:

1. Plotting polar curves
   • Define function as r[t_] (see above), then use: PolarPlot[r[t], {t, tmin, tmax}]
     
   • Plotting polar curve, θ = a
   • Convert to cartesian equation
     
2. Plot two polar curves
   • Define function as r1[t_] and r2[t_] (see above), then use: PolarPlot[{r1[t],r2[t]}, {t, tmin, tmax}]
3. Find the intersection(s) of two polar curves
   • Define function as r1[t_] and r2[t_] (see above), then use: Solve[r1[t] == r2[t], t]
4. Plot a polar curve along with a cartesian equation
   • Define variable plot1 using: plot1 = PolarPlot[r1[t], {t, tmin, tmax}]
   • Define variable plot2 using: plot2 = Plot[f[x], {x, xmin, xmax}]
   • Use: Show[plot1, plot2]

Section 7.1:

1. Find the general solution to differential equation
   • Store y′ into variable eqn using: eqn = y'[t] == f[t, y] where f[t, y] is the differential equation
     Note: Make sure to use y[t] for every dependent variable
   • Solve for y using: y[t]/.DSolve[eqn, y[t], t]
2. Solve Initial-Value Problem
   • Store y′ into variable eqn using: eqn = y'[t] == f[t, y] where f[t, y] is the differential equation
     Note: Make sure to use y[t] for every dependent variable
   • Solve for y using: y[t]/.DSolve[{eqn, y[a]==b}, y[t], t] where y(a) = b is the initial condition

Section 7.2:

1. Plotting direction field for y′ = f(x, y)
   • VectorPlot[{1, f[x, y]}, {x, xmin, xmax}, {y, ymin, ymax}, Axes -> True]
     
2. Euler's method
   • See Project __
     

Section 7.3:

1. Solving Separable Equations
   • DSolve[y'[x] == f[x, y], y[x], x] where f[x, y] is the separable equation
     

Section 8.1:

1. Sequences
   • List first i terms given a_n, using: Table[a_n, {n, nmin, i}] where a_n is the nth term and i is the number of terms desired
     
2. Plot Sequences
   • ListPlot[Table[a_n, {n, nmin, i}]] where a_n is the nth term and i is the number of terms desired
     Note: You can use the option PlotRange -> {ymin, ymax} to keep the y-values of the plot constant
3. Evalaute the Limit of Sequences as n → ∞
   • Limit[a_n, n -> Infinity] where a_n is the nth term

Section 8.2:

1. Series
   • Evaluate the sum of the first i terms of a series given a_n, using: Sum[a_n, {n, 1, i}] where a_n is the nth term and i is the number of terms desired
     
2. Find the partial sums, S_i, of a series
   • Table[Sum[a_n, {n, 1, k}], {k, 1, i}] where a_n is the nth term and i is the number of sums desired
     Note: For decimal output, use //N after the Table function
3. Plot Partial Sums
   • ListPlot[Table[Sum[a_n, {n, 1, k}], {k, 1, i}]] where a_n is the nth term and i is the number of terms desired
     Note: You can use the option PlotRange -> {ymin, ymax} to keep the y-values of the plot constant

Section 8.7:

1. Maclaurin Series
   • Find the first i terms of the Maclaurin series for f(x), using: Normal[Series[f[x], {x, 0, i}]] where i is the number of terms desired
     
2. Taylor Series
   • Find the first i terms of the Taylor series for f(x), using: Normal[Series[f[x], {x, a, i}]] where i is the number of terms desired and the series is centered about x = a using: Normal[Series[f[x], {x, a, i}]] where i is the number of terms desired and the series is centered about x = a