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

Introductory Instructions:

- Download Mathematica
- For basic mathematica usage instructions; using mathematica for algebra; using mathematica for precalculus; using mathematica for calculus I:

Section 5.2

- Evaluating sums (e.g., sum of i from 1 to n)
`Sum[i, {i, 1, n}]`

- Evaluating Riemann sums
- Store integrand into f(x) using:
`f[x_]:=function`

- Store Δx into variable dx using:
`dx=(b-a)/n`

– where a is lower limit of integration and b is upper limit of integration - Find x
_{i}and store into variable xi using:`xi=expression`

- Evaluate the sum and store into sum variable:
`sum = Sum[f[xi]*dx,{i,1,n}]`

- Evaluate the limit of the sum as n → ∞:
`Limit[sum, n -> Infinity]`

- Store integrand into f(x) using:
- Graphing Riemann sums
- See Project 1:

- See Project 1:

Section 5.2 & 5.3:

- 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

- Define function as f[x_] (see above), then use:
- 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.

- Define function as f[x_] (see above), then use:

Section 5.4:

- 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

- Define function as f[t_] as function (see above), then use:

Section 5.7 & Appendix G:

- Partial Fractions
- Define rational function as f[x_] as function (see above), then use:
`Apart[[f[x]]]`

- Define rational function as f[x_] as function (see above), then use:
- 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.

- To solve the system, expr1 = a,
expr2 = b, expr3 = c,
etc, use:

Section 5.9:

- Approximate Integration
- See Project 1:

Section 5.10:

- Improper Integrals
- 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.

- Formal Method: Use Limit function
- 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.

- Formal Method: Use Limit function

- Infinite Intervals:

Section 6.1:

- 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)

- Define function as f[x_] and g[x_] (see above), then use:
- 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]`

- Define function as f[y_] and g[y_] (see above), then use:
- 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

- Define function as f[y_] and g[y_] (see above), then use:

Appendix H:

- 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

- Define function as r[t_] (see above), then use:
- Plot two polar curves
- Define function as r1[t_] and r2[t_] (see above), then use:
`PolarPlot[{r1[t],r2[t]}, {t, tmin, tmax}]`

- Define function as r1[t_] and r2[t_] (see above), then use:
- 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]`

- Define function as r1[t_] and r2[t_] (see above), then use:
- 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]`

- Define variable plot1 using:

Section 7.1:

- 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]`

- Store y′ into variable eqn using:
- 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

- Store y′ into variable eqn using:

Section 7.2:

- Plotting direction field for y′ = f(x, y)
`VectorPlot[{1, f[x, y]}, {x, xmin, xmax}, {y, ymin, ymax}, Axes -> True]`

- Euler's method
- See Project __

- See Project __

Section 7.3:

- Solving Separable Equations
`DSolve[y'[x] == f[x, y], y[x], x]`

where f[x, y] is the separable equation

Section 8.1:

- 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

- List first i terms given a
- 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

- Evalaute the Limit of Sequences as n → ∞
`Limit[a_n, n -> Infinity]`

where a_n is the nth term

Section 8.2:

- 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

- Evaluate the sum of the first i terms of a series given a
- 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

- 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:

- 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

- Find the first i terms of the Maclaurin series for f(x),
using:
- 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

- Find the first i terms of the Taylor series for f(x),
using: