 # Calculus for Biological Scientists

## AppendixBList of Symbols

Symbol Description Location
$$p_0$$ the initial value of a quantity $$p$$ in a DTDS Example 1.1.4
$$p_t$$ the value of a quantity $$p$$ after $$t$$ steps in a DTDS Example 1.1.4
$$f(x)$$ the output value associated with an input value of $$x$$ in the function relationship represented by $$f$$ Item 1
$$(f \circ g)(x)$$ $$f$$ composed with the function $$g\text{,}$$ or $$f(g(x))$$ Definition 1.2.7
$$f^{-1}(x)$$ the inverse function of an invertible function $$f(x)$$ Definition 1.2.9
$$AROC_{[a,b]}$$ the average rate of change between $$x$$ values $$a$$ and $$b$$ for a given function Definition 1.2.13
$$t_d$$ the doubling time for a given exponential growth model Assemblage
$$t_h$$ the half life for a given exponential decay model Assemblage
$$b^*$$ an equilibrium point for a DTDS whose dependent variable is $$b$$ Definition 1.8.3
$$\lim_{x \to a}f(x)$$ The limit of $$f(x)$$ as $$x$$ approaches $$a$$ Definition 2.1.2
$$\lim_{x \to a^-}f(x)$$ The limit of $$f(x)$$ as $$x$$ approaches $$a$$ from the left Definition 2.1.4
$$\lim_{x \to a^+}f(x)$$ The limit of $$f(x)$$ as $$x$$ approaches $$a$$ from the right Definition 2.1.4
$$f'(a)$$ The derivative of $$f(x)$$ at the $$x$$ value $$a$$ Definition 2.2.3
$$f'(x)$$ The derivative function of $$f(x)$$ with respect to the variable $$x$$ Definition 2.3.2
$$f''(x)$$ The second derivative function of $$f(x)$$ with respect to the variable $$x$$ Definition 2.4.4
$$\frac{dy}{dx}$$ The derivative function of $$y$$ with respect to the variable $$x$$ Paragraph
$$\frac{d}{dx}[f]$$ The derivative function of $$f$$ with respect to the variable $$x$$ Paragraph
$$\frac{d^2 f}{dx^2}$$ The second derivative function of $$f$$ with respect to the variable $$x$$ Paragraph
$$L(x)$$ The linear approximation of a function centered at an $$x$$ value Paragraph
$$Q(x)$$ The quadratic approximation of a function centered at an $$x$$ value Assemblage
$$\lim_{x \to \infty}f(x)$$ The limit of $$f(x)$$ as $$x$$ gets arbitrarily large Paragraph
$$f_\infty(x)$$ The leading behavior of $$f(x)$$ at $$\infty$$ Definition 3.7.1
$$f_{-\infty}(x)$$ The leading behavior of $$f(x)$$ at $$-\infty$$ Definition 3.7.3
$$\int f(x) dx$$ The indefinite integral of $$f(x)$$ with respect to the variable $$x$$ Assemblage
$$\sum_{k=1}^n f(k)$$ The sum of expressions of the form $$f(k)$$ as $$k$$ goes from $$1$$ to $$n$$ Paragraph
$$L_n$$ A left Riemann sum for a function on an interval using $$n$$ rectangles Paragraph
$$R_n$$ A right Riemann sum for a function on an interval using $$n$$ rectangles Paragraph
$$M_n$$ A middle Riemann sum for a function on an interval using $$n$$ rectangles Paragraph
$$\int_a^b f(x) dx$$ The definite integral of $$f(x)$$ as $$x$$ values range from $$a$$ to $$b$$ Definition 4.4.3
$$f_{\operatorname{AVG} [a,b]}$$ The average value of $$f(x)$$ on the interval $$[a,b]$$ Assemblage
$$\left. F(x) \right|_a^b$$ The function $$F(x)$$ evaluated from $$a$$ to $$b\text{:}$$ $$F(b) - F(a)$$ Paragraph
$$\frac{dy}{dx}\bigg\vert_{(x,y)}$$ The derivative of $$y$$ with respect to $$x$$ evaluated at the point $$(x,y)$$ Warm-Up 4.6.1
$$E_{\Delta t}$$ An approximate value of the solution to an initial value problem using Euler’s method with a step size of $$\Delta t$$ Paragraph