Calculus I

Let's see how Wolfram|Alpha can handle or help handle typical types of problems in first-semester calculus.

# Functions

## Plot a Function

Example: Plot the following function in the window [0,5] x [0,10]:

(1)
$$f(x) = x(x-2)(x-3)$$

Walpha Query: plot x(x-2)(x-3), x=0 to x=5, y=0 to y=10

Result: Walpha plots the given function in the given window, just as you would expect it to.

Implications: No big deal here. Walpha functions as a basic plotting device. Certainly, giving students an easy way to plot the graphs of functions is likely to help them better understand those functions.

## Piecewise Functions

Example: Plot the following piecewise function on [-1,2]:

(2)
\begin{align} f(x)=\begin{cases}-x & x\leq 0 \\ x^5 & x>0\end{cases} \end{align}

Walpha Query:

plot Piecewise[{{-x,x<= 0},{x^5,x>0}}] from x=-1 to x=2


Syntax: Without the word "plot", and no "for" clause, the expression will yield a plot, with bounds determined by Walpha. Changing "plot" to "integral" will compute the definite integral, if the "for" clause is present, or the indefinite integral otherwise (no "Show steps" option). Derivatives are handled by asking for "derivative" instead of "plot", but the steps shown are hard to interpret.

# Limits

## Calculate a Limit

Example: Find

(3)
\begin{align} \lim_{x\to 2} \frac{x^2-4}{x-2} \end{align}

Walpha Query: limit as x->2 of (x^2-4)/(x-2)

Example: Find

(4)
\begin{align} \lim_{x\to 2} \frac{\sqrt{x+4}-2}{x} \end{align}

Walpha Query: limit as x->2 of (sqrt(x+4)-2)/x

Show Steps: Clicking on "Show steps" will demonstrate a way to obtain the given limit. In the first example above, the numerator is factored, while in the second example, L'Hospital's Rule is used. In other examples, the steps may include L'Hospital's Rule, and steps are generally detailed, explicitly stating that constant, sum, product, and quotient rules are used, as well as continuity. Sometimes the "steps" will be a single line with just the numerical value, for example in Squeeze Theorem sorts of problems (x*sin(1/x)).

Additional Info: Walpha also includes a plot of the function, centered (on the x-axis) at the limit point, and a series expansion for the function. When series expansions appear, an option to show "More terms" may be present.

Implications: Walpha is likely to show students L'Hospital's Rule before students have learned about derivatives. L'Hospital's Rule is likely to be used, even when students will likely be expected to use algebraic techniques. Similarly, Walpha will show students series expansions of functions before they have been seen in class.

Syntax: Seems to be vaguely flexible. Instead of "limit as x->a for f(x)", you can leave out "as" or "for" (but not both). The "as x->a" bit may appear after the function, in which case "limit for" does not even need to be present.

## Calculate One-Side Limits

Example: Find

(5)
\begin{align} \lim_{x\to 0^+} \lfloor x\rfloor,\quad \lim_{x\to 0^-} \lfloor x\rfloor \end{align}

Walpha Query: limit as x->0 for floor(x)

Syntax: There doesn't seem to be special syntax for specifying a one-sided limit. Entering the two-sided limit, though, will answer the question. If the two-sided limit exists, it is the same as both one-sided limits. If this is not the case, Walpha seems to automatically try to find both one-sided limits. Limits to vertical asymptotes are computed by Walpha.

Additional Info: A plot will be included.

## Limits to +/- Infinity

Example: Find

(6)
\begin{align} \lim_{x\to\infty} \frac{x^2+2x+\frac{1}{x}}{1000x-2x^2} \end{align}

Walpha Query: (x^2+2x+1/x)/(1000x-2x^2) as x->infinity

Show Steps: Seems to work as for usual limits.

Additional Info: A series expansion at infinity is given, but no plot.

# Derivatives

## Find the Derivative of a Polynomial

Example: Find

(7)
\begin{align} \frac{d}{dx} \left ( 4x^3 - 7x^2 + x + 2) \end{align}

Walpha Query: derivative 4x^3-7x^2+x+2

Show Steps: Clicking on "show steps" will demonstrate how the Addition, Subtraction, Constant Multiple, Constant, and Power Rules are used to find this derivative, although the rules are not named as such.

Additional Info: Walpha also plots the derivative, tells you it's a parabola, finds its roots, finds its global minimum, and (strangely) computes the definite integral using the two roots as lower and upper bounds.

Implications: Walpha easily solves this type of problem. Instructors asking their students to show all their steps will find that Walpha provides that information, too, although it doesn't name the various differentiation rules used in the solution. Students struggling with this kind of problem might find the "show steps" information very useful in understanding how those rules are applied, however.

Plotting the derivative is nice, but plotting it on the same axes as the original function would be more instructive for students, I think. Finding the roots of the derivative would save students time in applications like optimization and curve-sketching, especially since the graph of the derivative is provided, as well.

## Find the Derivative of a Rational Function

Example: Find

(8)
\begin{align} \frac{d}{dx} \left ( \frac{4x^2}{x^2-x-6} \right ) \end{align}

Walpha Query: derivative 4x^2/(x^2 - x - 6)

Show Steps: As with the derivative of a polynomial, Walpha shows how the Addition, Subtraction, Constant Multiple, Constant, and Power Rules are applied, although they are not named as such. Walpha does name the Quotient Rule, although it's not the usual Quotient Rule. Instead, the following version is used:

(9)
\begin{align} \frac{d}{dx} \left ( \frac{u}{v} \right ) = \frac{ \frac{du}{dx} }{v} - \frac{u \frac{dv}{dx} }{v^2} \end{align}

Additional Info: Walpha also plots the derivative (although not with a window that shows all of its important features) and provides an alternate form for the derivative (as a single fraction). Walpha also provides series expansions of the derivative at 0 and $\infty$. (Users can request additional terms of these series be provided.) It also provides the limits of the derivative at infinity, effectively providing the horizontal asymptotes of the function.

Implications: As with the derivative of a polynomial, Walpha easily solves this type of problem. Instructors asking their students to show all their steps will find that Walpha provides that information, too, although it doesn't name the various differentiation rules used in the solution and uses a variant Quotient Rule. Students struggling with this kind of problem might find the "show steps" information very useful in understanding how those rules are applied, however, although the variant Quotient Rule would likely confuse some students.

The plot of the derivative isn't very useful in this case since key features are not clearly shown. In particular, students learning to "curve sketch" won't find these plots very instructional. Having the series expansions of the derivative handy is neat but not particularly useful for any application. Likewise for the limits at infinity. These calculations are more useful for the original function than for the derivative. Their presence might even confuse some students.

# Applications of Derivatives

## Optimize a Function

Example: Find the global maximum of the following function:

(10)
$$f(x)=x(300-2x)$$

Walpha Query: maximize x(300-2x)

Results: Walpha returns the global maximum of 11,250 at x=75, as you would expect it to. "Show steps" isn't an option for this result, however. Walpha also plots the function in a couple of different windows with the global maximum noted by a red dot.

Example: Find the maximum value of the following function on the interval $[-1, 5]$:

(11)
$$f(x) = -x^3+4x^2$$

W|A Query: Maximize -x^3+4x^2 on [-1, 5]

Results: W|A returns the maximum value of $\frac{256}{27}$ at $x= \frac{8}{3}$, as expected. There's no "Show steps" option. One can get a decimal approximation of these two values by clicking "Approximate form." W|A also provides a plot of the function on the interval in question with the maximum value marked by a red dot.

Implications: If the goal is to have students simply find the maximum of a function, then W|A provides this readily. Since it doesn't show its steps, however, (a) students whose instructors as them to show their work will have to work through the problem anyway and (b) students interested in learning how maxima are computed won't learn anything about that.

Having the plot handy is useful for students since there's a temptation when solving these kinds of problems to simply take the derivative, set it equal to zero, solve, and go with whatever answer results from that. Checking to see if the critical point is actually an extremum of the correct kind, particularly when bounds on the independent variable are given, is important, and the plot allows students to make that check. If the goal is to have students use sign changes in the derivative to make that check, however, that goal will be circumvented by easy access to the plot.

page revision: 14, last edited: 17 Jun 2009 04:11