12.4 Derivatives (Page 8/18)

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Find the equation of a tangent line to the curve of the function $f (x) = 5 x^{2} - x + 4$ at $x = 2.$

$y = 19 x - 16$

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Finding the instantaneous speed of a particle

If a function measures position versus time, the derivative measures displacement versus time, or the speed of the object. A change in speed or direction relative to a change in time is known as velocity . The velocity at a given instant is known as instantaneous velocity .

In trying to find the speed or velocity of an object at a given instant, we seem to encounter a contradiction. We normally define speed as the distance traveled divided by the elapsed time. But in an instant, no distance is traveled, and no time elapses. How will we divide zero by zero? The use of a derivative solves this problem. A derivative allows us to say that even while the object’s velocity is constantly changing, it has a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance.

A General Note

Instantaneous velocity

Let the function $s (t)$ represent the position of an object at time $t .$ The instantaneous velocity or velocity of the object at time $t = a$ is given by

$s^{'} (a) = \lim_{h \to 0} \frac{s (a + h) - s (a)}{h}$

Finding the instantaneous velocity

A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after $t$ seconds is given by $s (t) = −16 t^{2} + 36 t + 200,$ find the instantaneous velocity of the ball at $t = 2.$

First, we must find the derivative $s^{'} (t)$ . Then we evaluate the derivative at $t = 2,$ using $s (a + h) = - 16 {(a + h)}^{2} + 36 (a + h) + 200$ and $s (a) = - 16 a^{2} + 36 a + 200.$

$\begin{array}{l} s^{'} (a) = \lim_{h \to 0} \frac{s (a + h) - s (a)}{h} \\ = \lim_{h \to 0} \frac{- 16 {(a + h)}^{2} + 36 (a + h) + 200 - (- 16 a^{2} + 36 a + 200)}{h} \\ = \lim_{h \to 0} \frac{- 16 (a^{2} + 2 a h + h^{2}) + 36 (a + h) + 200 - (- 16 a^{2} + 36 a + 200)}{h} \\ = \lim_{h \to 0} \frac{- 16 a^{2} - 32 a h - 16 h^{2} + 36 a + 36 h + 200 + 16 a^{2} - 36 a - 200}{h} \\ = \lim_{h \to 0} \frac{- 16 a^{2} - 32 a h - 16 h^{2} + 36 a + 36 h + 200 + 16 a^{2} - 36 a - 200}{h} \\ = \lim_{h \to 0} \frac{- 32 a h - 16 h^{2} + 36 h}{h} \\ = \lim_{h \to 0} \frac{h (- 32 a - 16 h + 36)}{h} \\ = \lim_{h \to 0} (- 32 a - 16 h + 36) \\ = - 32 a - 16 \cdot 0 + 36 \\ s^{'} (a) = - 32 a + 36 \\ s^{'} (2) = - 32 (2) + 36 \\ = - 28 \end{array}$

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Try It

A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after $t$ seconds is given by $s = - 16 t^{2} + 60 t - 12.$ What is its instantaneous velocity after 4 seconds?

–68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.

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Media

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Key equations

average rate of change	$AROC = \frac{f (a + h) - f (a)}{h}$
derivative of a function	$f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$

Key concepts

The slope of the secant line connecting two points is the average rate of change of the function between those points. See [link] .
The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See [link] , [link] , and [link] .
The difference quotient is the quotient in the formula for the instantaneous rate of change:
$\frac{f (a + h) - f (a)}{h}$
Instantaneous rates of change can be used to find solutions to many real-world problems. See [link] .
The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See [link] .
Instantaneous rates of change can be interpreted to describe real-world situations. See [link] and [link] .
Some functions are not differentiable at a point or points. See [link] .
The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See [link] .
Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time.
Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See [link] .

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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