Cos : The cosine of an angle $0\leq\alpha\leq 180^{\circ}$ or $\cos \alpha$ is defined by finding a right triangle with angle $\alpha$ and dividing the length of the leg adjacent to $\alpha$ by the length of the hypotenuse. For arbitrary angles the cosine function can be extended in a periodic way by inscribing the right triangle in a circle. The cosine can be used to calculate unknown side lengths in a triangle via the cosine law and in the case of right triangles via its definition. The Pythagorean theorem implies that $\cos^2 \alpha+\sin^2 \alpha=1.$

Derivative : A derivative of $f$ also written as $f^{\prime}$ or $\frac{d}{dx}f$ or $\frac{df}{dx}$ is defined by $f^{\prime}(x)=\lim \limits_{h\to 0}\frac{f(x+h)-f(x)}{h}.$ The derivative $f^{\prime}(x)$ can be interpreted as the speed at which the function value is changing or as the slope of the tangent in the graph of $f$ at $x$. For example the derivative of a constant function is 0, the derivative of $f(x)=x$ is 1 and the derivative of $x^2$ is $2x.$ All local extrema of differentiable functions $f$ are roots of the derivative of $f.$

Factorial : The factorial of a positive integer $n$ written as $n!$ is defined by $n!=1\cdot2\cdot3\ldots\cdot n.$ For example we have $5!=1\cdot2\cdot3\cdot4\cdot 5=120.$ $n!$ is the number of different permutations of $n$ objects.

Linear Taylor approximation : The linear Taylor approximation at $x_0$ of a differentiable function $f$ is given by: $f(x_0)+f^{\prime}(x_0)(x-x_0).$

Geometric mean : The geometric mean of numbers $x_1, x_2,\ldots, x_N$ is defined as $\sqrt[N]{x_1\cdot x_2\cdot \ldots\cdot x_N}.$ For example the geometric mean of 1 and 2 is $\sqrt{2}.$ The inequality between geometric mean and arithmetic mean states the the arithmetic mean is always as least as big as the geometric mean.

Inequality : An inequality is a statement that contains an inequality sign between two expressions.

Number : A number $x$ is a mathematical symbol representing a quantity.

Product : A product is the result of a multiplication.

Square root : The square root of $x$ denoted by $\sqrt{x}$ is the positive number such that $(\sqrt{x})^2=\sqrt{x}\cdot \sqrt{x}=x.$ For example $\sqrt{9}=3.$

Squared : $x$ squared refers to the number $x^2=x\cdot x.$ For example 3 squared equals 9.

Limit : The limit of a function $f$ for $x$ converging to $x_0$ or $\lim\limits_{x\to 0}f(x)$ is a number $y$ such that for every $\epsilon\gt 0$ there is a $\delta\gt 0$ with $|f(x)-y|\lt\epsilon$ for all $|x-x_0|\lt\delta.$ This means that if $x$ only gets close enough to $x_0$ it will get and stay arbitrarily close to $y.$

Numerator : The numerator of a fraction $\frac{a}{b}$ is the number $a$ above the fraction bar.

Polynomial expansion : Polynomial expansion refers to multiplying out brackets in a factorization of a polynomial. The most commonly used polynomial expansions are $(x+y)^2=x^2+2xy+y^2,$ $(x-y)^2=x^2-2xy+y^2$ and $(x+y)(x-y)=x^2-y^2.$

Squared : $x$ squared refers to the number $x^2=x\cdot x.$ For example 3 squared equals 9.