Modeling Rates of Change: Introduction to the Issues

Modeling Rates of Change: Introduction to the Issues

The Legacy of Galileo, Newton, and Leibniz Galileo Galilei (1564-1642) was interested in falling bodies. He forged a new scientific methodology: observe nature, eperiment to test what you observe, and construct theories that eplain the observations.

The Legacy of Galileo, Newton, and Leibniz Galileo (1564-1642): Eperiment, then draw conclusions. Sir Isaac Newton (1642-1727) using his new tools of calculus, eplained mathematically why an object, falling under the influence of gravity, will have constant acceleration of 9.8m/sec 2. His laws of motion unified Newton s laws of falling bodies, Kepler s laws of planetary motion, the motion of a simple pendulum, and virtually every other instance of dynamic motion observed in the universe.

The Legacy of Galileo, Newton, and Leibniz Galileo (1564-1642): Eperiment, then draw conclusions. Newton (1642-1727): Invented/used calculus to eplain motion Gottfried Wilhelm Leibniz (1646-1716) independently co-invented calculus, taking a slightly different point of view ( infinitesimal calculus ) but also studied rates of change in a general setting. We take a lot of our notation from Leibniz.

The Legacy of Galileo, Newton, and Leibniz Newton s Question: How do we find the velocity of a moving object at time t? What in fact do we mean by velocity of the object at the instant of time t?

The Legacy of Galileo, Newton, and Leibniz Newton s Question: How do we find the velocity of a moving object at time t? What in fact do we mean by velocity of the object at the instant of time t? It s straightforward to find the average velocity of an object during a time interval [t 1, t 2 ]: average velocity = change in position change in time = y t.

The Legacy of Galileo, Newton, and Leibniz Newton s Question: How do we find the velocity of a moving object at time t? What in fact do we mean by velocity of the object at the instant of time t? It s straightforward to find the average velocity of an object during a time interval [t 1, t 2 ]: change in position average velocity = = y change in time t. But what is meant by instantaneous velocity?

Drop a ball from the top of a building... At time t, how far has the ball fallen?

Drop a ball from the top of a building... At time t, how far has the ball fallen? Measure it! time (s) distance (m) 0.10 0.049 0.20 0.196 0.30 0.441 0.40 0.784 0.50 1.225 0.60 1.764 0.70 2.401 0.80 3.136 0.90 3.969 5 4 3 2 1 1.00 4.900 0.2 0.4 0.6 0.8 1

Drop a ball from the top of a building... At time t, how far has the ball fallen? Measure it! time (s) distance (m) 0.10 0.049 0.20 0.196 0.30 0.441 0.40 0.784 0.50 1.225 0.60 1.764 0.70 2.401 0.80 3.136 0.90 3.969 5 4 3 2 1 How fast is the ball falling at time t? 1.00 4.900 0.2 0.4 0.6 0.8 1

Drop a ball from the top of a building... At time t, how far has the ball fallen? Measure it! time (s) distance (m) 0.10 0.049 0.20 0.196 0.30 0.441 0.40 0.784 0.50 1.225 0.60 1.764 0.70 2.401 0.80 3.136 0.90 3.969 5 4 3 2 1 1.00 4.900 0.2 0.4 0.6 0.8 1 How fast is the ball falling at time t? A little trickier...

Average Speed 5 Definition The average velocity from t = t 1 to t = t 2 is 4 3 avg velocity = change in distance change in time 2 1 0.2 0.4 0.6 0.8 1

Average Speed 5 Definition The average velocity from t = t 1 to t = t 2 is change in distance avg velocity = change in time = slope of secant line 4 3 2 1 0.2 0.4 0.6 0.8 1

Average Speed 5 Definition The average velocity from t = t 1 to t = t 2 is change in distance avg velocity = change in time = slope of secant line 4 3 2 1 0.2 0.4 0.6 0.8 1

Average Speed 5 Definition The average velocity from t = t 1 to t = t 2 is change in distance avg velocity = change in time = slope of secant line 4 3 2 1 0.2 0.4 0.6 0.8 1

Average Speed 5 Definition The average velocity from t = t 1 to t = t 2 is change in distance avg velocity = change in time = slope of secant line 4 3 2 1 0.2 0.4 0.6 0.8 1

Average Speed 5 Definition The average velocity from t = t 1 to t = t 2 is change in distance avg velocity = change in time = slope of secant line 4 3 2 1 0.2 0.4 0.6 0.8 1

Plot position f versus time t: y = f(t)

Plot position f versus time t: y = f(t) f(b) f(a) a b Pick two points on the curve (a, f(a)) and (b, f(b)).

Plot position f versus time t: y = f(t) f(a+h) f(a) h a a+h Pick two points on the curve (a, f(a)) and (b, f(b)). Rewrite b = a + h.

Plot position f versus time t: y = f(t) f(a+h) f(a) h a a+h Pick two points on the curve (a, f(a)) and (b, f(b)). Rewrite b = a + h. Slope of the line connecting them: avg velocity = m = f(a + h) f(a) h difference quotient

Plot position f versus time t: y = f(t) f(a) f(a+h) h a a+h Pick two points on the curve (a, f(a)) and (b, f(b)). Rewrite b = a + h. Slope of the line connecting them: avg velocity = m = f(a + h) f(a) h difference quotient

Plot position f versus time t: y = f(t) f(a) f(a+h) h a a+h Pick two points on the curve (a, f(a)) and (b, f(b)). Rewrite b = a + h. Slope of the line connecting them: avg velocity = m = f(a + h) f(a) h difference quotient

Plot position f versus time t: y = f(t) f(a) f(a+h) h a a+h Pick two points on the curve (a, f(a)) and (b, f(b)). Rewrite b = a + h. Slope of the line connecting them: avg velocity = m = f(a + h) f(a) h The smaller h is, the more useful m is! difference quotient

Goal: Rates of change in general Think: f() is distance versus time, or profit versus production volume, or birthrate versus population, or...

Goal: Rates of change in general Think: f() is distance versus time, or profit versus production volume, or birthrate versus population, or... Definition Given a function f, the average rate of change of f over an interval [, + h] is f( + h) f(). h The average rate of change is also what we have called the difference quotient over the interval.

Goal: Rates of change in general Think: f() is distance versus time, or profit versus production volume, or birthrate versus population, or... Definition Given a function f, the average rate of change of f over an interval [, + h] is f( + h) f(). h The average rate of change is also what we have called the difference quotient over the interval. Definition The instantaneous rate of change of a function at a point is the limit of the average rates of change over intervals [, + h] as h 0.

Average rate of change Instantaneous rate of change f(a+h) f(a) h a a+h

Average rate of change Instantaneous rate of change f(a) f(a+h) h a a+h

Average rate of change Instantaneous rate of change f(a) f(a+h) h a a+h

Future goals: 1. Get good at limits. 2. Eplore instantaneous rates of change further, as limits of difference quotients. 3. Eplore the geometric meaning of the definition of instantaneous rate of change at a point. 4. Apply the definition to each of the elementary functions to see if there are formula-like rules for calculating the instantaneous rate of change. 5. Use the definition of instantaneous rate of change and its consequences to obtain eplicit functions for the position, velocity, and acceleration of a falling object.

Future goals: 1. Get good at limits. 2. Eplore instantaneous rates of change further, as limits of difference quotients. 3. Eplore the geometric meaning of the definition of instantaneous rate of change at a point. 4. Apply the definition to each of the elementary functions to see if there are formula-like rules for calculating the instantaneous rate of change. 5. Use the definition of instantaneous rate of change and its consequences to obtain eplicit functions for the position, velocity, and acceleration of a falling object.

Limit of a Function Definition We say that a function f approaches the limit L as approaches a, written lim f() = L, a if we can make f() as close to L as we want by taking sufficiently close to a.

Limit of a Function Definition We say that a function f approaches the limit L as approaches a, written lim f() = L, a if we can make f() as close to L as we want by taking sufficiently close to a.

Limit of a Function Definition We say that a function f approaches the limit L as approaches a, written lim f() = L, a if we can make f() as close to L as we want by taking sufficiently close to a. L a

Limit of a Function Definition We say that a function f approaches the limit L as approaches a, written lim f() = L, a if we can make f() as close to L as we want by taking sufficiently close to a. f() L a

Limit of a Function Definition We say that a function f approaches the limit L as approaches a, written lim f() = L, a if we can make f() as close to L as we want by taking sufficiently close to a. f() L Δy Δ a

Limit of a Function Definition We say that a function f approaches the limit L as approaches a, written lim f() = L, a if we can make f() as close to L as we want by taking sufficiently close to a. f() L Δy Δ i.e. If you need y to be smaller, you only need to make smaller ( means change ) a

One-sided limits f(a) L r L l Right-handed limit: L r = lim a + f() if f() gets closer to L r as gets closer to a from the right a Left-handed limit: L l = lim a f() if f() gets closer to L l as gets closer to a from the left

One-sided limits f(a) L r L l Right-handed limit: L r = lim a + f() if f() gets closer to L r as gets closer to a from the right a Left-handed limit: L l = lim a f() if f() gets closer to L l as gets closer to a from the left Theorem The limit of f as a eists if and only if both the right-hand and left-hand limits eist and have the same value, i.e. lim f() = L if and only if lim a f() = L and lim a f() = L. a +

Eamples FIGURE 9 lim 2 2 + 3 8 a (a) lim ƒ=l (b) lim ƒ=l a _ 2 a + 1 1 ƒ By comparing lim Definition l with the lim definitions of one-sided limits, we see that lowing is true. 1 1 0 8 8 0 4-4 4 3 4 lim f L l a -4 4 4 if and only FIGURE if lim f L 9 la -4 4 and lim(a) flim ƒ= L la a _ Compute y= 1 2 3 4 5-4 -8-4 -8-8 The graph of a function t is shown in Figure 10. Use it to state th V EXAMPLE 7 (if they eist) of the following: (a) lim t (b) lim t (c) l 2 (d) lim t (e) lim t (f) l 5 for the following function: l 2 l 5 SOLUTION From the graph we see that the values of t approach 3 as approaches the left, but they y approach 1 as approaches 2 from Vthe EXAMPLE right. Therefore 7 The graph 4 (a) lim (if they eist) of the follow t 3 and (b) lim t 1 l 2 l 2 3 (a) lim t (b) (c) Since the left and right limits y= l 2 are different, we conclude from (3) that lim l 2 does not eist. (d) lim t (e) l 5 The graph 1also shows that (d) lim t 2 and (e) SOLUTION lim t From 2the graph w 0 l 5 l 5 1 2 3 4 5 the left, but they approach -4 lim t l 2 lim t l 5 By comparing Definiti lowing is true. 3 lim f L l a

Eamples FIGURE 9 lim 2 2 + 3 = 0 8 a (a) lim ƒ=l (b) lim ƒ=l a _ lim 2 a + 1 1 1 = 2 lim 1 ƒ By comparing Definition l with the definitions is of undefined one-sided limits, we see that lowing is true. 0 8 8 0 4-4 4 3 4 lim f L l a -4 4 4 if and only FIGURE if lim f L 9 la -4 4 and lim(a) flim ƒ= L la a _ Compute y= 1 2 3 4 5-4 -8-4 -8-8 The graph of a function t is shown in Figure 10. Use it to state th V EXAMPLE 7 (if they eist) of the following: (a) lim t (b) lim t (c) l 2 (d) lim t (e) lim t (f) l 5 for the following function: l 2 l 5 SOLUTION From the graph we see that the values of t approach 3 as approaches the left, but they y approach 1 as approaches 2 from Vthe EXAMPLE right. Therefore 7 The graph 4 (a) lim (if they eist) of the follow t 3 and (b) lim t 1 l 2 l 2 3 (a) lim t (b) (c) Since the left and right limits y= l 2 are different, we conclude from (3) that lim l 2 does not eist. (d) lim t (e) l 5 The graph 1also shows that (d) lim t 2 and (e) SOLUTION lim t From 2the graph w 0 l 5 l 5 1 2 3 4 5 the left, but they approach -4 lim t l 2 lim t l 5 By comparing Definiti lowing is true. 3 lim f L l a

Theorem If lim a f() = A and lim a g() = B both eist, then 1. lim a (f() + g()) = lim a f() + lim a g() = A + B 2. lim a (f() g()) = lim a f() lim a g() = A B 3. lim a (f()g()) = lim a f() lim a g() = A B 4. If B 0, then lim a (f()/g()) = lim a f()/ lim a g() = A/B. In short: to take a limit Step 1: Can you just plug in? If so, do it. Step 2: If not, is there some sort of algebraic manipulation (like cancellation) that can be done to fi the problem? If so, do it. Then plug in. Step 3: Learn some special limit to fi common problems. (Later) If in doubt, graph it!

Eamples 2 1. lim 2 + 3 2 1 2. lim 1 1 3. lim 0 + 2 2

Eamples 2 2 1. lim = 0 because if f() = 2 + 3 +3, then f(2) = 0. 2 1 2. lim 1 1 3. lim 0 + 2 2

Eamples 2 1. lim 2 + 3 2 1 2. lim 1 1 2 = 0 because if f() = +3, then f(2) = 0. 0 0 3. lim 0 + 2 2

Eamples 2 1. lim 2 + 3 2 1 2. lim 1 1 2 = 0 because if f() = +3, then f(2) = 0. 0 0 If f() = 2 1 1, then f() is undefined at = 1. 3. lim 0 + 2 2

Eamples 2 1. lim 2 + 3 2 1 2. lim 1 1 2 = 0 because if f() = +3, then f(2) = 0. 0 0 If f() = 2 1 1, then f() is undefined at = 1. However, so long as 1, f() = 2 1 1 = ( + 1)( 1) 1 = + 1. 3. lim 0 + 2 2

Eamples 2 1. lim 2 + 3 2 1 2. lim 1 1 2 = 0 because if f() = +3, then f(2) = 0. 0 0 If f() = 2 1 1, then f() is undefined at = 1. However, so long as 1, So 3. lim 0 f() = 2 1 1 = ( + 1)( 1) 1 = + 1. 2 1 lim 1 1 = lim + 1 = 1 + 1 = 2. 1 + 2 2

Eamples 2 1. lim 2 + 3 2 1 2. lim 1 1 2 = 0 because if f() = +3, then f(2) = 0. 0 0 If f() = 2 1 1, then f() is undefined at = 1. However, so long as 1, So 3. lim 0 f() = 2 1 1 = ( + 1)( 1) 1 = + 1. 2 1 lim 1 1 = lim + 1 = 1 + 1 = 2. 1 + 2 2 0 0, so again, f() is undefined at a.

Eamples 3. lim 0 + 2 2 0 0, so again, f() is undefined at a. Multiply top and bottom by the conjugate: ( ) ( ) + 2 2 + 2 2 + 2 + 2 lim = lim 0 0 + 2 + 2

Eamples 3. lim 0 + 2 2 0 0, so again, f() is undefined at a. Multiply top and bottom by the conjugate: ( ) ( ) + 2 2 + 2 2 + 2 + 2 lim = lim 0 0 + 2 + 2 = lim 0 + 2 2 ( + 2 + 2) since (a b)(a + b) = a 2 b 2

Eamples 3. lim 0 + 2 2 0 0, so again, f() is undefined at a. Multiply top and bottom by the conjugate: ( ) ( ) + 2 2 + 2 2 + 2 + 2 lim = lim 0 0 + 2 + 2 + 2 2 = lim 0 ( + 2 + 2) = lim 0 ( + 2 + 2) since (a b)(a + b) = a 2 b 2

Eamples 3. lim 0 + 2 2 0 0, so again, f() is undefined at a. Multiply top and bottom by the conjugate: ( ) ( ) + 2 2 + 2 2 + 2 + 2 lim = lim 0 0 + 2 + 2 + 2 2 = lim 0 ( + 2 + 2) = lim 0 ( + 2 + 2) 1 = lim 0 + 2 + 2 since (a b)(a + b) = a 2 b 2

Eamples 3. lim 0 + 2 2 0 0, so again, f() is undefined at a. Multiply top and bottom by the conjugate: ( ) ( ) + 2 2 + 2 2 + 2 + 2 lim = lim 0 0 + 2 + 2 + 2 2 = lim 0 ( + 2 + 2) = lim 0 ( + 2 + 2) 1 = lim = 1 0 + 2 + 2 2 2 since (a b)(a + b) = a 2 b 2

You try: 1. lim 1 2 3+2 2 +4 5 2. lim 2 3. lim 0 4. lim 0 (3 + ) 2 3 2

You try: 1. lim 1 2 3+2 2 +4 5 = 1 6 2 2. lim 2 = 1 3. lim 0 is undefined 1-2 -1 1 2-1 -2 4. lim 0 (3 + ) 2 3 2 = 6

Badly behaved eample: f() = csc(1/) -0.5 0.5 Zoom way in: -0.005 0.005 (denser and denser vertical asymptotes) lim csc(1/) does not eist, and lim 0 + csc(1/) does not eist 0