AP Calculus AB

What people get wrong

1. Plugging in the target value and stopping when you get 0/0 instead of recognizing it as a signal to factor, rationalize, or combine fractions before trying again.
2. Treating the limit value and the function value f(c) as automatically the same thing — at a removable discontinuity they differ, and the limit exists independently of what f(c) is defined to be.
3. Checking only that one-sided limits are equal and forgetting the third continuity condition, that this common value must also equal f(c), which is exactly what makes removable discontinuities easy to miss.
4. Getting the sign of an infinite limit wrong by not separately checking the sign of the expression as x approaches from the left versus the right of a vertical asymptote.

What people get wrong

1. Plugging h=0 directly into the difference quotient before simplifying, which produces 0/0 and stalls the problem. Always algebraically simplify (expand, factor, or rationalize) to cancel the h in the denominator first.
2. Assuming continuity guarantees differentiability. Continuity is necessary but not sufficient — functions like $|x|$ or $x^{1/3}$ are continuous but not differentiable at x=0. Always separately check for corners, cusps, or vertical tangents.
3. Reversing the order of terms in the quotient rule numerator, giving $fg'-f'g$ instead of $f'g-fg'$. Memorize it as 'low d-high minus high d-low' (using the denominator as 'low') to keep the sign correct.
4. Treating the derivative of $a^x$ the same as $e^x$ and forgetting the $\ln a$ factor, or forgetting the extra $\ln a$ in the denominator for $\log_a x$. Only $e^x$ and $\ln x$ are 'clean'; other bases pick up a $\ln a$ factor.

What people get wrong

1. Forgetting to multiply by the derivative of the inner function in a composition (e.g., writing $\frac{d}{dx}[\sin(x^3)]=\cos(x^3)$ instead of $3x^2\cos(x^3)$). Always ask 'what is the inner function, and what is its derivative?' before finishing.
2. Differentiating a mixed term like $xy^2$ implicitly by treating it as a single variable and forgetting the product rule. Write it out as $\frac{d}{dx}[x]\cdot y^2+x\cdot\frac{d}{dx}[y^2]$ first, then apply the chain rule to the $y^2$ piece.
3. Mixing up the derivatives of arcsin and arccos, or arctan and arccot — they differ only by a sign. Memorize arcsin/arctan/arcsec as positive and their co-function partners as the negatives of those same expressions.
4. Leaving $\frac{dy}{dx}$ unsimplified inside a second-derivative expression instead of substituting the first-derivative formula back in. The AP rubric expects the final answer in terms of x and y (or a number), not in terms of $y'$.

What people get wrong

1. Substituting the numerical values into the related-rates equation before differentiating. Differentiate the general equation with respect to t first, and only plug in the specific instant's numbers afterward.
2. Treating speed and velocity as the same thing on a motion problem. Speed is $|v(t)|$; a particle can have decreasing speed while its velocity is becoming more negative. Always check the sign of both v and a.
3. Computing total distance as $\int v(t)\,dt$ (which gives net displacement) instead of $\int|v(t)|\,dt$. Find where v(t)=0 first, then add the absolute values of displacement on each piece.
4. Applying L'Hospital's rule to a limit that isn't actually 0/0 or ∞/∞. Plug in the limiting value first to confirm the indeterminate form; if it isn't indeterminate, just evaluate directly.

What people get wrong

1. Applying the Mean Value Theorem or Rolle's Theorem without checking both hypotheses — instead, explicitly verify continuity on the closed interval and differentiability on the open interval (watch for corners or vertical tangents) before invoking the conclusion.
2. Treating every critical number as a local extremum — instead, always confirm with a sign change in f' (first derivative test) or the sign of f'' (second derivative test); some critical numbers are neither, like a horizontal inflection point.
3. Making sign errors when building a sign chart for f' or f'' — instead, factor the derivative completely, test one value in each subinterval, and track the sign of each factor separately before multiplying them.
4. Assuming the second derivative test always works — instead, when $f''(c)=0$, recognize the test is inconclusive and fall back on the first derivative test.

What people get wrong

1. Forgetting the chain-rule factor in FTC Part 1 when the bound is a function of x other than x itself — always multiply by the derivative of that bound, not just plug it in.
2. Forgetting to flip the sign when the variable is the lower limit of an accumulation function — rewrite ∫ from x to a as −∫ from a to x before differentiating.
3. Dropping the +C on indefinite integral answers — it costs points on free-response even though it cancels out in definite integrals.
4. Using u-substitution on a definite integral but evaluating the u-antiderivative at the original x-bounds — convert the limits to u-values, or convert the antiderivative back to x before plugging in the original bounds.

What people get wrong

1. Forgetting to add the constant of integration when separating variables, or forgetting to solve for it using the initial condition — always write '+C' immediately after integrating, then substitute the given point before doing anything else.
2. Dropping the absolute value when integrating $\int \frac{1}{y}dy = \ln|y|+C$, then mishandling the sign when exponentiating — carry the absolute value through and let the constant $A=\pm e^{C}$ absorb the sign, determined later by the initial condition.
3. Treating $\frac{dy}{dx}=ky$ as if it produces a linear function $y=kx+C$ instead of the exponential solution $y=Ce^{kt}$ — memorize that this specific equation always gives exponential, not linear, growth.
4. Verifying a solution by only checking one point instead of confirming the equation holds identically for all x — differentiate symbolically and simplify both sides completely.

What people get wrong

1. Computing $\int_a^b f(x)\,dx$ and calling it the average value — you must also divide by $(b-a)$; write the formula down every time before plugging in.
2. Finding total distance by just taking $\left|\int_a^b v(t)\,dt\right|$ — this only gives displacement's magnitude. Instead find every zero of $v(t)$ in $[a,b]$, integrate on each subinterval, and add the absolute values.
3. Setting up a washer as $\pi\int (R-r)^2\,dx$ instead of $\pi\int (R^2-r^2)\,dx$ — these are algebraically different. Always square the two radii separately, then subtract.
4. Forgetting to shift the radius when rotating about a line other than the x- or y-axis (e.g. using $R(x)=\sqrt{x}$ instead of $\sqrt{x}+1$ when the axis is $y=-1$). Always measure radius as distance from the axis of rotation to the curve.