Common Mistakes

For \(R=\{(1,1),(2,2)\}\) on \(A=\{1,2,3\}\), \(R\) is not reflexive because \((3,3)\) is missing.

In testing transitivity, \((1,2)\) and \((2,1)\) form a chain that requires \((1,1)\).

Onto depends on the codomain: \(e^x\) is not onto \(\mathbb{R}\) because negative numbers and 0 are never outputs.

Showing two particular inputs give different outputs proves nothing; you must start from f(x₁) = f(x₂) in general.

\(\cos^{-1}(-\tfrac12)\neq-\tfrac\pi3\). The range of \(\cos^{-1}\) is \([0,\pi]\), so the answer is \(\pi-\tfrac\pi3=\tfrac{2\pi}3\).

\(\sin^{-1}x\neq\dfrac{1}{\sin x}\). \(\sin^{-1}\) is an angle; \(\frac1{\sin x}=\csc x\).

\(\tan^{-1}(\tan\frac{3\pi}4)=-\frac\pi4\), not \(\frac{3\pi}4\): the answer must lie in \((-\frac\pi2,\frac\pi2)\).

Writing \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}^2 = \begin{bmatrix} 4 & 1 \\ 0 & 9 \end{bmatrix}\).

Applying the number identity to matrices.

\(AB=O\) does not force \(A=O\) or \(B=O\), and \((A+B)^2=A^2+AB+BA+B^2\), not \(A^2+2AB+B^2\), unless \(AB=BA\).

Writing \((AB)^T = A^TB^T\).

Swapping the diagonal entries but forgetting to change the signs of the off-diagonal entries (or vice versa).

For a 3 × 3 matrix with \(|A| = 4\), claiming \(|2A| = 8\).

Expanding \(a_{11}M_{11} + a_{12}M_{12} + a_{13}M_{13}\) with all plus signs.

For a \(3\times3\) matrix, \(|2A|=2^3|A|=8|A|\), not \(2|A|\). A scalar multiplies every row.

Writing “Area = −12 square units”.

\(A_{ij}=(-1)^{i+j}M_{ij}\). The sign pattern is a chessboard starting with + in the top-left corner.

Taking the cofactor matrix itself as adj A.

Multiplying the constants column on the wrong side.

\(\frac{d}{dx}\sin(3x)=3\cos(3x)\), not \(\cos(3x)\).

\(\frac{d}{dx}(y^3)=3y^2\frac{dy}{dx}\), and \(\frac{d}{dx}(xy)=y+x\frac{dy}{dx}\).

\(\frac{d}{dx}x^x\neq x\cdot x^{x-1}\) and \(\neq x^x\log x\). Correct: \(x^x(1+\log x)\).

\(\frac{d^2y}{dx^2}\neq\frac{d^2y/dt^2}{d^2x/dt^2}\). Differentiate \(\frac{dy}{dx}\) with respect to \(t\), then divide by \(\frac{dx}{dt}\).

Differentiate \(V=\frac43\pi r^3\) first, then put \(r=5\). If you substitute first, \(V\) becomes a constant with derivative 0.

"Decreasing at 2 cm/s" means \(\frac{dx}{dt}=-2\).

The absolute maximum of \(x^3-3x\) on \([0,3]\) is at the end point \(x=3\) (value 18), not at a critical point.

A local maximum need not be the largest value overall; a local minimum value can even exceed a local maximum value.

\(\int x\sin x\,dx\neq\int x\,dx\cdot\int\sin x\,dx\). Products need substitution or by parts.

\(\int x^{-1}dx=\log|x|+C\), but \(\int x^{-2}dx=-x^{-1}+C\) – the log rule is only for the power −1.

After substituting u = g(x) in a definite integral, either change the limits to u-values or substitute back before using the original limits.

In \(\int x e^x dx\), taking \(u=e^x\) makes the integral harder. ILATE says take \(u=x\) (algebraic before exponential).

\(\int_0^{2\pi}\sin x\,dx=0\), but the area between \(\sin x\) and the axis on \([0,2\pi]\) is \(2+2=4\).

When using horizontal strips, the limits must be y-values.

For \(\left(1+(y')^2\right)^{3/2}=y''\), square both sides first: the degree is 2, not \(\frac23\) or undefined.

After \(\log|y|=x^2+C\), write \(y=Ae^{x^2}\) – the constant multiplies, it does not add.

For \(x\frac{dy}{dx}+2y=x^2\), divide by \(x\) first: \(P=\frac2x\), \(\text{IF}=x^2\) – not \(e^{\int2\,dx}\).

For ratio \(m:n\) from \(A\) to \(B\), the weight \(m\) goes with \(\vec b\) (the far point).

Projection of \(\vec a\) on \(\vec b\) divides by \(|\vec b|\), not \(|\vec a|\).

\(\hat j\times\hat i=-\hat k\), not \(\hat k\). Reversing the order changes the sign.

\(2,-1,2\) are direction ratios; the direction cosines are \(\frac23,-\frac13,\frac23\).

In \(\frac{1-x}{2}=\frac{y}{3}=\frac{2z+1}{4}\), rewrite as \(\frac{x-1}{-2}=\frac{y}{3}=\frac{z+\frac12}{2}\): DRs are \(-2,3,2\).

“At least” means ≥ and “at most” means ≤. Reversing one inequality changes the whole feasible region.

On an unbounded region the largest value in the table need not be a maximum. Draw \(px+qy>M\) and check whether it meets the region.

\(P(A\mid B)\) divides by \(P(B)\); \(P(B\mid A)\) divides by \(P(A)\). They are equal only when \(P(A)=P(B)\).

Mutually exclusive events with positive probabilities cannot be independent: \(P(A\cap B)=0\) but \(P(A)P(B)>0\).