Calculus - Santowski 11/17/2015 Calculus - Santowski 1 B.1.1 – Limit Definition of the Derivative

Calculus - Santowski

04/20/23Calculus - Santowski1

B.1.1 – Limit Definition of the Derivative

Lesson Objectives


1. Introduce limit definitions of a derivative as an instantaneous rate of change at a point

2. Calculate the derivative of simple polynomial and rational functions from first principles

3. Calculate the derivative of simple polynomial and rational functions using the TI-89

4. Calculate derivatives and apply to real world scenarios

Fast Five


1. Expand (2 + h)3

2. Find limx9(x2 - 81)/(x - 9)3. Give an eqn of a fcn that has a jump

discontinuity4. Simplify the quotient [f(7+h)-f(7)]/h if

f(x) = (x + 2)0.5

5. Graph y1(x) = x2 and y2(x) = |x|. Continually zoom in at x = 0. Explain what “local linearity” means. How might this idea be relevant given our last 2 lessons?

(A) Review


We have determined a way to estimate an instantaneous rate of change (or the slope of a tangent line) which is done by means of a series of secant lines such that the secant slope is very close to the tangent slope. We accomplish this "closeness" by simply moving our secant point closer and closer to our tangency point, such that the secant line almost sits on top of the tangent line because the secant point is almost on top of our tangency point.

Q? Is there an algebraic method that we can use to simplify the tedious approach of calculating secant slopes and get right to the tangent slope??

(B) Review - Notations


We have several notations for tangent slopes

We will add one more

where x or h represent the difference between our secant point and our tangent point as seen in our diagram on the next slide

€

limΔx →0

Δy

Δx= lim

x2 →x1

y2 − y1

x2 − x1

= limx →a

f (x) − f (a)

x − a

€

limΔx →0

f (x + Δx) − f (x)

(x + Δx) − (x)= lim

h →0

f (x + h) − f (x)

(x + h) − (x)= lim

h →0

f (x + h) − f (x)

h

(B) Notations - Graph


(C) Notations - Derivatives


This special limit of

is the keystone of differential calculus, so we assign it a special name and a notation.

We will call this fundamental limit the derivative of a function f(x) at a point x = a

The notation is f `(a) and is read as f prime of a

€

limh →0

f (a + h) − f (a)

h

(C) Notations - Derivatives


Alternative notations for the derivative are :

f `(x) y`

which means the derivative at a point, x = a

KEY POINT: In all our work with the derivative at a point, please remember its two interpretations: (1) the slope of the tangent line drawn at a specified x value, and (2) the instantaneous rate of change at a specified point.

€

dy

dx x= a

€

dy

dx

(D) Using the Derivative to Determine the Slope of a Tangent Line


Determine the slope of the tangent line to f(x) = -2x2 + x - 5 at x = 1

Alternate way to ask the same question:

Determine the instantaneous rate of change of f(x) = -2x2 + x - 5 at x = 1

Determine the value of the derivative of f(x) = -2x2 + x - 5 at x = 1

Determine f ‘ (1) if f(x) = -2x2 + x - 5

Evaluate

€

dy

dx x=1

if y = −2x 2 + x − 5



€

m = ′ f (1) = limh →0

f (1+ h) − f (1)

h

m = ′ f (1) = limh →0

−2 1+ h( )2

+ 1+ h( ) − 5( ) − −6( )

h

m = ′ f (1) = limh →0

−2 h2 + 2h +1( ) + 1+ h( ) − 5( ) − −6( )

h

m = ′ f (1) = limh →0

−2h2 − 3h − 6( ) − −6( )

h

m = ′ f (1) = limh →0

/ h −2h − 3( )/ h

m = ′ f (1) = limh →0

−2 / h − 3( )

m = ′ f (1) = −3



Now let’s confirm this using the graphing features and the calculus features of the TI-89

So ask the calculator to graph and determine the tangent eqn:


Now do it from the homescreen several different ways:

(1) define your function and then use the limit definition of the derivative

04/20/23 Calculus - Santowski 12


Use the nDeriv command (ask the calculator to calculate the numerical value of the derivative of your function at x = 1

Ask the GDC to find dy/dx at x = 1

04/20/23 Calculus - Santowski 13



A football is kicked and its height is modeled by the equation h(t) = -4.9t² + 16t + 1, where h is height measured in meters and t is time in seconds. Determine the instantaneous rate of change of height at 1s, 2s, 3s.

ANSWER: So the slope of the tangent line (or the instantaneous rate of change of height) is 6.2 so, in context, the rate of change of a distance is called a speed (or velocity), which in this case would be 6.2 m/s at t = 1 sec. now simply repeat, but use t = 2,3 rather than 1

(D) Using the Derivative to Determine the Slope of a Tangent LineSoln is:


€

m =dh

dt t=1

= limΔt →0

h(1+ Δt) − h(1)

Δt

m =dh

dt t=1

= limΔt →0

−4.9 1+ Δt( )2

+16 1+ Δt( ) +1( ) − −4.9 1( )2

+16 1( ) +1( )

Δt

m =dh

dt t=1

= limΔt →0

−4.9Δt 2 + 6.2Δt +12.1( ) − 12.1( )

Δt

m =dh

dt t=1

= limΔt →0

−4.9Δt + 6.2( )

m =dh

dt t=1

= 6.2



A business estimates its profit function by the formula P(x) = x3 - 2x + 2 where x is millions of units produced and P(x) is in billions of dollars. Determine the value of the derivative at x = ½ and at x = 1½. How would you interpret these derivative values?



For the following functions, determine the value of the derivative at the given x co-ordinate

€

g(x) =1

2x +1 determine ′ g (−1)

h(t) = t + 2 determine dh

dt t= 7



Here is an example of a B LEVEL mathematical application question:

Determine the point(s) on f(x) = x3 – 12x where the function has a horizontal tangent line(s).

(E) Interpretation of Derivatives


What follows in the next slides are various questions which involve interpretations of derivatives what do they really mean in the context of “word problems”

Here you are expected to verbally convey your understanding of derivatives along the lines of rates of change, rate functions, etc...

Most of the following “applications” are independent of algebra and graphs



Ex. 1. The cost C in dollars of building a house of A square feet in area is given by the function C = f(A). What is the practical interpretation of the function dC/dA or f `(A)? One option on interpreting is to consider units (dollars per square foot).

Ex. 2 You are told that water is flowing through a pipe at a constant rate of 10 litres per second. Interpret this rate as the derivative of some function.



Ex. 3. If q = f(p) gives the number of pounds of sugar produced when the price per pound is p dollars, then what are the units and meaning of f `(3) = 50?

Ex. 4. The number of bacteria after t hours in a controlled experiment is n = f(t). What is the meaning of f `(4)? Suppose that there is an unlimited amount of space and nutrients for the bacteria. Which is larger f `(4) or f `(8)? If the supply of space and nutrients were limited, would that affect your conclusion?



Ex. 5. The fuel consumption (measured in litres per hour) of a car traveling at a speed of v km/hr is c = f(v). What is the meaning of f `(v)? What are its units? Write a sentence that explains the meaning of the equation f `(30) = -0.05.

Ex. 6. The quantity (in meters) of a certain fabric that is sold by a manufacturer at a price of p dollars per meter is Q = f(p). What is the meaning of f `(16). Its units are? Is f `(16) positive or negative? Explain.



Ex. 7. A company budgets for research and development for a new product. Let m represent the amount of money invested in R&D and T be the time until the product is ready to market.

(A) Give reasonable units for T and m. What is T ' in these units.

(B) What is the economic interpretation of T '?

(C) Would you expect T ' to be positive or negative? Explain?



Ex. 8 Interpret each sentence as a statement about a function and its derivative. In each case, clearly indicate what the function is, what each variable means and appropriate units. Make a sketch of a graph that best reflects the context.

(A) The price of a product decreases as more of it is produced.

(B) The increase in demand for a new product decreases over time.

(C) The work force is growing more slowly than it was five years ago.

(D) Health care costs continue to rise but at a higher rate than 4 years ago.

(E) During the past 2 years, Canada has continued to cut its consumption of imported oil

(L) Homework


Page 191-194

(1) C LEVEL: Q5-9, (graphs) (2) C LEVEL: Q1,17-22 (algebraic work –

find f’(a) - verify using GDC)(3) B LEVEL: Q45-57 odds (word

problems)(4) B LEVEL: Q38-44(5) A LEVEL: WORKSHEET (p115): Q1-16

odds

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Calculus - Santowski 11/17/2015 Calculus - Santowski 1 B.1.1 – Limit Definition of the Derivative