A4 - Using Limits to Find Instantaneous Rates of Change - Derivatives at a Point

IB Math HL&SL - Santowski

(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) Notations and Definitions

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

lim lim( ) ( )

lim( ) ( )

x x x x a

y

x

f x f x

x x

f x f a

x a

0

2 1

2 12 1

lim( ) ( )

( ) ( )lim

( ) ( )

( ) ( )

x h

f x x f x

x x x

f x h f x

x h x

0 0

(B) Notations and Definitions- 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 = aThe notation is f `(a) and is read as f prime of a

We can further generalize the formula wherein we do not specify a value for x as:

f x

f x h f x

hh( ) lim

( ) ( )0

lim( ) ( )

h

f a h f a

h

0

(C) Notations - Derivatives

Alternative notations for the derivative are :f `(x) y`dy/dx

which means the derivative at a specific 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

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

ex 1. Determine the equation of the tangent line to the curve f(x) = -x2 + 3x - 5 at the point (-4,-33)

Then the equation of the line becomes y = 11x + b and we find b to equal (-33) = 11(-4) + b so b is 1 so y = 11x + 11

11

11lim

11lim

335312168lim

)33(5434lim

)4()4(lim

0

2

0

2

0

2

0

0

m

hmh

hhm

h

hhhm

h

hhm

h

fhfm

h

h

h

h

h

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

mh t t h t

t

mt t h

t

mt t

tm t

m

m

t

t

t

t

lim( ) ( )

lim[ . ( ) ( ) ] ( )

lim[ . . . ] .

lim ( . . )

. ( ) .

.

0

0

2

0

2

0

4 9 1 1 6 1 1 1

4 9 6 2 1 2 1 1 2 1

4 9 6 2

4 9 0 6 2

6 2

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

ex 3. 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?

(E) Homework

Stewart, 1989, Calculus – A First Course, Chap 2.1, p78, 4,5,6,9