Unit 4: Contextual Applications of Differentiation

19 terms. Say what each one means, then open the card to check.

• Instantaneous rate of change

  The derivative $f'(a)$: how fast $f$ is changing at the single input $a$. It is the slope of the tangent line to the graph at that point.

  Example. If $B(t)$ is the number of bees in a hive after $t$ days and $B'(12) = 40$, the hive is gaining about 40 bees per day on day 12.

• Units of a derivative

  The units of the output divided by the units of the input. A second derivative divides by the input units twice.

  Example. If $D(h)$ is distance in miles after $h$ hours, $D'(h)$ is in miles per hour and $D''(h)$ is in miles per hour per hour.

• Average rate of change

  $\frac{f(b) - f(a)}{b - a}$: the change in output per unit of input over a whole interval, which is the slope of the secant line.

  Example. A kite at 20 meters when t = 1 second and at 35 meters when t = 6 seconds rose an average of 3 meters per second.

• Estimating a derivative from a table

  Divide the change in output by the change in input across the two listed inputs closest to the point, one on each side of it.

  Example. If $C(8) = 31$ and $C(12) = 39$, then $C'(10) \approx \frac{39 - 31}{12 - 8} = 2$.

• Interpreting a derivative in context

  A full reading names the quantity, the moment, the rate with its units, and whether the quantity is rising or falling.

  Example. If $S(t)$ is a store's stock in boxes on day $t$, $S'(3) = -12$ means that on day 3 the stock is falling by 12 boxes per day.

• Position function

  A function, such as $x(t)$ or $s(t)$, that gives an object's location on a line at time $t$, measured from a fixed origin.

  Example. $s(t) = 4t - t^2$ meters puts an object 3 meters to the right of the origin at $t = 1$.

• Velocity

  The derivative of position, $v(t) = x'(t)$. Its sign gives the direction: positive for right or up, negative for left or down.

  Example. For $x(t) = t^2 - 4t$, $v(1) = -2$, so at $t = 1$ the object moves left at 2 units per second.

• Speed

  The absolute value of velocity, $|v(t)|$. Speed has no direction and is never negative.

  Example. Velocities of 5 and -5 meters per second both mean a speed of 5 meters per second.

• Acceleration

  The derivative of velocity, $a(t) = v'(t) = x''(t)$: how fast the velocity is changing.

  Example. If $v(t) = 3t^2$ feet per second, then $a(2) = 12$ feet per second per second.

• Speeding up and slowing down

  Speed increases when velocity and acceleration have the same sign, and decreases when their signs are opposite.

  Example. An object with velocity -3 and acceleration -1 is speeding up, even though its acceleration is negative.

• Change of direction

  A particle changes direction where its velocity changes sign, not at every point where its velocity is 0.

  Example. $v(t) = (t - 2)^2$ is 0 at $t = 2$ but never negative, so the particle does not turn around there.

• At rest

  A particle is at rest at an instant when its velocity is 0.

  Example. For $v(t) = t^2 - 9$ with $t \geq 0$, the particle is at rest only at $t = 3$.

• Related rates

  Rates of change of quantities that are linked by an equation. Differentiating the equation with respect to time links their rates.

  Example. A circle's circumference $C = 2\pi r$ gives $\frac{dC}{dt} = 2\pi \frac{dr}{dt}$, so the circumference grows $2\pi$ times as fast as the radius.

• Differentiating with respect to time

  Every changing quantity is a function of $t$, so the chain rule gives each one its own rate factor, such as $\frac{dx}{dt}$.

  Example. Differentiating $x^2 + y^2 = 100$ with respect to $t$ gives $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$.

• Values at the instant

  Numbers that are true only at the moment asked about. Substitute them after differentiating, or a changing quantity is treated as a constant.

  Example. If a side is 4 cm at one moment, keep it as $s$, differentiate, and only then put in $s = 4$.

• Local linear approximation

  Estimating $f(x)$ near $x = a$ with the tangent line $L(x) = f(a) + f'(a)(x - a)$.

  Example. With $g(1) = 5$ and $g'(1) = 2$, $g(1.1) \approx 5 + 2(0.1) = 5.2$.

• Overestimate or underestimate

  A tangent line lies below a concave-up graph, so it underestimates, and above a concave-down graph, so it overestimates.

  Example. $\sqrt{x}$ is concave down, so its tangent line at $x = 9$ gives a value slightly above $\sqrt{9.5}$.

• Indeterminate form

  A limit that substitutes to $\frac{0}{0}$ or $\frac{\infty}{\infty}$. The form alone does not decide the value; more work is needed.

  Example. $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ substitutes to $\frac{0}{0}$, yet the limit is 4.

• L’Hôpital’s rule

  If $\frac{f(x)}{g(x)}$ has the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, with $f$ and $g$ differentiable and $g'(x) \neq 0$ near the point, its limit equals $\lim \frac{f'(x)}{g'(x)}$ when that limit exists or is infinite.

  Example. $\lim_{x \to 0} \frac{e^x - 1}{x}$ has the form $\frac{0}{0}$, and $\frac{e^x}{1} \to 1$, so the limit is 1.