Intro to First Derivative

Mr. Simonds’ MTH 251

I n t r o d u c t i o n t o t h e F i r s t D e r i v a t i v e | 1

According to simplified Newtonian physics, if a projectile is fired from a height of 10 m at an initial speed of 49 m/s, then the elevation of the object (measured in m) is given by the function ( ) 24.9 49 10g t t t= − + + where t is the amount of time (measured in s) that has passed since the

projectile was launched.

We define the average velocity of the projectile over the time interval [ ]1 2,t t to be ( ) ( )2 1

2 1

g t g tt t−−

.

Let’s find a generalized formula for the average velocity over the interval [ ]1 2,t t .


2 | I n t r o d u c t i o n t o t h e F i r s t D e r i v a t i v e

When analyzing rates of change, we sometimes find it advantageous to specify the initial time and the length of the time interval rather than the initial and final times of the interval. In this situation, we tend to identify the initial time as t and the length of the time interval as h. It should be noted that technically h could be negative; in that case, t would actually be the end of the time interval and h would be the length of the time interval.

Let’s find a formula for the average velocity (using the model on page 1) in terms of t and h. ( ) 24.9 49 10g t t t= − + +



Let’s use the formula ( ) 24.9 49 10g t t t= − + + to determine the average velocity of the projectile

over the interval [ ]4,7 and let’s include units while making the calculation. Let’s then draw a

graphical representation of the average velocity onto Figure 1. Let’s use both of our average velocity formulas to confirm the average velocity of the projectile over [ ]4,7 . Let’s do this in a unit-less context.

Let’s use both of our average velocity formulas to help us determine the (instantaneous) velocity of the projectile exactly four seconds into its flight. Let’s do this in a unit-less context.

Figure 1: g



Let’s discuss three fundamental differences between the difference quotient and the first derivative function.

Definitions

The difference quotient for the function ( )y f x= is the expression ( ) ( )f x h f x

h+ −

.

The first derivative function for f is ( ) ( ) ( )0

limh

f x h f xf x

h→

+ −′ = .

Figure 2: secant line to ( )cosy t= Figure 3: tangent line to ( )cosy t=



Let’s use both of the definitions of the first derivative to find the first derivative formula for the function ( ) 3h x x= .

Alternate Definition

The first derivative function for ( )y f x= is ( ) ( ) ( )limt x

f t f xf x

t x→

−′ =

−.



Let’s use the alternate definition of the first derivative function to find a formula for ( )g t′ where

( ) 1 2g t t= − + .



The function ( ) 13

xw xx+

=−

is shown in Figure 4. Let’s use the standard definition of the first

derivative function to help us determine the equation of the tangent line to w at 1.

Figure 4: w

1y =

3x =



Stanly and Hooch are filling a large pot with water using the spigot at the back of their house. Stanly and Hooch are only 6 and 8, so they haven’t yet figured out that they don’t need to hold the pot whilst it fills. Because the pot is really large and their arms not so much so, at times more water is sloshing out of the pot than is flowing in from the spigot. The rate at which the volume of water in the pot is changing (measured in gal/min) is given by the formula ( ) ( ) ( )5cos 3sin 2R t t t= − + where t is the amount of time (min) that has elapsed since

the boys began to fill the pot. It can be shown that ( ) ( ) ( )5sin 3cosR t t t′ = − − . Let’s approximate the values of ( )3.7R and

( )3.7R′ to the nearest 10th (including their units) and then write sentences that convey the

meaning of the values in the context of this problem.

Intro to First Derivative

Documents

Transcript of Intro to First Derivative