A Preview of Calculus: practice problems

Key terms · 8

average velocity

the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval $[t, a]$ (if $t < a$ or $[a, t]$ if $t > a)$, with a position given by $s (t),$ that is $v_{\text{ave}} = \frac{s (t) - s (a)}{t - a}$

differential calculus

the field of calculus concerned with the study of derivatives and their applications

instantaneous velocity

The instantaneous velocity of an object with a position function that is given by $s (t)$ is the value that the average velocities on intervals of the form $[t, a]$ and $[a, t]$ approach as the values of t move closer to $a,$ provided such a value exists

integral calculus

the study of integrals and their applications

limit

the process of letting x or t approach a in an expression; the limit of a function $f (x)$ as x approaches a is the value that $f (x)$ approaches as x approaches a

multivariable calculus

the study of the calculus of functions of two or more variables

secant

A secant line to a function $f (x)$ at a is a line through the point $(a, f (a))$ and another point on the function; the slope of the secant line is given by $m_{\sec} = \frac{f (x) - f (a)}{x - a}$

tangent

A tangent line to the graph of a function at a point $(a, f (a))$ is the line that secant lines through $(a, f (a))$ approach as they are taken through points on the function with x-values that approach a; the slope of the tangent line to a graph at a measures the rate of change of the function at a