Instructor guide

• Estimate a limit from a graph or a table of values, from both sides.
• Explain why the value f(a) plays no part in the limit.
• Recognize three ways a limit can fail: a jump, unbounded growth, and oscillation.

Load the hole with a stray point, f(1) = 4. Ask the class to vote on the limit before anyone moves the point. Then switch to sin(π/x) and show how the table at powers of 10 gives 0 every time.

Students who answer with f(a). Students who trust a table without checking both sides.

• State the sum, product, quotient, power and root laws, with their conditions.
• Decide when a law cannot be used, and explain why that does not settle the limit.
• Use direct substitution for polynomials and rational functions.

Use "Try: neither limit exists". Ask whether the sum can have a limit when neither piece does, and collect predictions before revealing it.

Writing −2/0 as a value. Concluding that a limit does not exist because a law failed.

• Recognize the 0/0 form and connect it to a shared factor through the factor theorem.
• Evaluate limits of rational functions by factoring and cancelling.
• Explain why cancelling is valid inside a limit.

Show the hole, then turn on the simplified function. Ask what the two graphs disagree about and why the limit cannot tell them apart.

Writing 0/0 = 1. Stopping after one cancellation when a factor of x − a remains.

• Use the conjugate to remove a square root from a 0/0 quotient.
• Simplify (A − B)(A + B) = A² − B² correctly, keeping track of signs.

Before any algebra, have students estimate the first limit from the table. Then ask why multiplying by (√x + 2)/(√x + 2) cannot change the value.

Multiplying only the numerator. Splitting √(x + 5) into √x + √5. Losing the factor of −1 after 4 − (x + 3).

• Compute one-sided limits and use them to decide whether a two-sided limit exists.
• Rewrite an absolute value as two cases and evaluate limits of each piece.
• Distinguish a jump, an infinite limit, and a limit that exists.

Show |x − 2|/(x − 2) with "Show the one-sided limits" off. Ask pairs to predict both one-sided limits, then reveal.

Dropping the absolute value bars. Treating ∞ as a number the limit equals.

• State the Squeeze Theorem and check both of its conditions.
• Build bounds from −1 ≤ sin(·) ≤ 1, including the |x| case.
• Explain the limit of sin x / x.

Turn off "Show f" and shrink the window. Ask whether the class can still name the limit. Then load sin(1/x) and ask what changed.

Multiplying an inequality by x when x may be negative. Saying sin(1/x) → 0.

• Evaluate limits of rational functions as x → ±∞ by dividing by the highest power.
• Use degrees to find horizontal asymptotes.
• Handle √(x²) = |x| at −∞.

Ask for a guess of f(1,000,000) for the first function, then zoom out together. Finish with the square-root example, where the two ends differ.

Writing ∞/∞ = 1. Ignoring the sign of odd powers at −∞.

• Check the three conditions for continuity at a point.
• Classify removable, jump and infinite discontinuities.
• Apply the Intermediate Value Theorem, and use bisection to locate a root.

Use the "choose k" function and ask the class to find k before anyone touches the slider. In the bisection graph, choose 1/x and ask why bisection converges to a point that is not a root.

Assuming defined means continuous. Using the IVT without checking continuity.

• Interpret the ε-δ definition as an input and output tolerance.
• Find δ as a function of ε for linear limits, and for x² using min(1, ε/5).
• Explain why the order of the quantifiers matters.

Run "Divide ε by 4" as a class challenge: one student sets ε, another must find a δ. Then use "Lock δ" to show why a single δ cannot work for every ε.

Choosing δ = ε/4 for x² at 2. Thinking δ must be the largest value that works.

• Compute secant slopes and explain the tangent slope as their limit.
• Evaluate f′(a) from the limit definition.
• Recognize where a derivative fails to exist.

Keep "Show the tangent line" off and ask the class to predict f′(1) for x² from the table. Then reveal it. Load |x| at 0 and compare the two sides.

Substituting h = 0 before simplifying. Forgetting that h can be negative.

• Sketch f′ from the graph of f.
• Connect peaks and valleys of f with zeros of f′.
• Identify corners and vertical tangents where f′ does not exist.

Before sweeping, have students sketch f′ for x³ − 3x on paper. Then sweep and compare.

Copying the height of f instead of its slope. Placing the zeros of f′ at the zeros of f.

• Apply the power, product and quotient rules.
• Explain the product rule with the area of a growing rectangle.

Show the rectangle with a large Δx and ask which of the three new pieces should matter as Δx shrinks. Then shrink it.

Writing (fg)′ = f′g′. Reversing the order in the quotient rule.

• Identify the inside and outside functions of a composition.
• Apply the chain rule, including hidden compositions such as sin³x.

Use the number lines: ask how much the x step stretches on each link, and what that means for the whole chain.

Forgetting the inside derivative. Changing the inside when differentiating the outside.

• Differentiate equations in x and y implicitly.
• Find tangent slopes at points on curves that are not functions.
• Locate horizontal and vertical tangents from the slope formula.

On the circle, ask where the tangent is vertical before anyone drags the point. Connect the answer to the denominator of −x/y.

Dropping dy/dx from y terms. Forgetting the product rule on xy.

• Explain why neither the power rule nor the exponential rule applies to a variable power.
• Differentiate functions such as xˣ by taking logs and differentiating implicitly.
• Use log laws to differentiate long products and quotients.

Ask the class to vote on the derivative of xˣ before opening the page. Then tick each rule alone, and finally both, and ask why the two wrong answers add up to the right one.

Writing ln(xˣ) as (ln x)ˣ. Splitting ln(x + 1). Stopping at y′/y without multiplying by y.

• Compute dy = f′(x) dx and explain its meaning on the tangent line.
• Compare dy with the actual change Δy, and explain why they agree for small dx.
• Estimate absolute and relative error in a computed quantity from a measurement error.

Load the volume of a ball and set dx to the measurement error. Ask how big the error in volume is before anyone looks at dy.

Confusing dy with Δy. Reporting relative error without dividing by the quantity itself.

• Write the linearization L(x) = f(a) + f′(a)(x − a).
• Choose a good base point and use L to estimate values.
• Describe how the error shrinks as x approaches a.

Ask students to estimate √4.1 to three decimal places without a calculator, in pairs. Then show the graph and the error table.

Using a base point that cannot be evaluated exactly. Using the slope at x instead of at a.

• Set up an equation linking changing quantities.
• Differentiate with respect to time and solve for an unknown rate.

Before pressing Play on the ladder, ask whether the top falls at a constant speed. Sketch predictions of the rate graph, then play.

Substituting values before differentiating. Ignoring the sign of a rate.

• Use the signs of f′ and f″ to find intervals of increase and concavity.
• Find and classify critical numbers and inflection points.

Hide f″ and ask where the inflection points of f must be, using only the graph of f′.

Assuming every critical number is an extremum. Assuming f″ = 0 always gives an inflection point.

• Translate a word problem into a function of one variable using the constraint.
• Find and justify the maximum or minimum, checking the domain.

Run the box as a competition: students try to find the best cut by dragging, then the class compares with the calculus answer.

Differentiating before using the constraint. Reporting the critical number instead of the optimal value.

• State the Mean Value Theorem and check its hypotheses.
• Find values of c that satisfy its conclusion.
• Explain what can go wrong when a hypothesis fails.

Use the driving example: ask whether a speed of exactly 80 km/h was unavoidable, then show |x| and 1/x, where the conclusion fails.

Giving a c outside (a, b). Skipping the hypotheses.

• Compute left, right and midpoint sums by hand.
• Predict whether a sum overestimates or underestimates from the shape of the graph.
• Explain why rectangles below the axis count as negative.

Before showing the error column, ask which method should be most accurate for x². Then compare all four with n = 4 and double n twice.

Using n + 1 sample points. Forgetting to multiply by Δx. Choosing the wrong endpoints for a midpoint sum.

• Write Δx and the sample point x_i for a right sum on [0, b].
• Use summation formulas to write a Riemann sum as a formula in n.
• Evaluate the definite integral as the limit of that formula.

Set n to 4, then use "Multiply n by 10" four times. Ask students to predict the limit from the readout before revealing the formula in n.

Treating the limit as a finished sum with n = 100000. Dropping the 1/n³ factor in the sum of squares.

• Explain why the derivative of an area function is the integrand.
• Evaluate definite integrals with antiderivatives.
• Differentiate integrals with variable upper limits, including the chain rule.

Open with the puzzle on the board: derivatives are slopes, integrals are areas, and the two undo each other, so slope and area must be opposites. Let students argue about it, then sweep slowly with cos t and pause where A(x) has a peak.

Subtracting in the wrong order. Losing a sign with −cos x.

• Explain why antiderivatives differ by a constant, using the graph of the family.
• Integrate powers, exponentials and trigonometric functions term by term.
• Rewrite roots and quotients as powers before integrating, and use an initial value to find C.

Turn on "Show the whole family" and ask why every curve has the same slope at the same x. Then ask which one passes through a given point.

Forgetting + C. Integrating a quotient by integrating the top and bottom separately. Using the power rule with n = −1.

• Choose u so that du appears in the integrand, up to a constant.
• Rewrite and evaluate indefinite integrals by substitution.
• Change the limits of a definite integral to u-values, or substitute back before evaluating.

Turn on "Show how the limits move" and drag the upper limit. Ask why the two shaded areas stay equal even though the regions look nothing alike.

Leaving some x in the integral after substituting. Keeping the old limits after changing to u.

• Distinguish net area (the integral) from total area.
• Find the zeros in an interval and split the integral there.
• Use geometry to evaluate integrals of semicircles and triangles.

Load sin x on a full period and ask for the integral before showing it. Then check "Flip the regions below the axis" and compare.

Reporting a net area of 0 as "no area". Missing a zero inside the interval.

• Find the intersections of two curves and use them as limits.
• Decide which curve is on top by testing a point.
• Split the integral where the curves cross, and set up top minus bottom.

Load y = x³ and y = x and drag the strip across the middle crossing. Ask what happens if you integrate x − x³ over all of [−1, 1].

Subtracting bottom minus top. Integrating across a crossing in one piece.

• Compute the average value of a function on an interval.
• Find every c in the interval where f(c) equals the average.
• Connect the theorem to the Mean Value Theorem for derivatives through the Fundamental Theorem.

Ask students to set the level by eye for sin x on [0, π] before pressing "Let it settle". Compare their guesses with 2/π.

Averaging only the endpoint values. Forgetting to divide by b − a. Keeping a value of c outside the interval.