F(x) approaches . f(x) approaches

[Pages:10]Section 15.1 ? Limits

Goals: 1. To use limit theorems to evaluate the limits of polynomial functions. 2. To use limit theorems to evaluate the limits of certain trig functions.

I. An Introduction to Limits Suppose you were to sketch the graph of the function f given by f (x) = x2 - 3x + 2 , x 2 . For

x-2 all values other than x = 2 , you can use standard methods of graphing. However, at x = 2 , it is not clear what to expect. To get an idea of the behavior we will develop three methods.

A. Numerically

x approaches 2 from the left

x approaches 2 from the right

x

2

f(x)

?

f(x) approaches _______.

f(x) approaches _______.

B. Graphically When you graph the function on your calculator, what does it appear to be doing?

C. Limit Notation 1. lim f (x) = _______

x2

2. Informal description of a limit ? If f (x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f (x) , as x approaches c, is L.

lim f (x) = L

xc

D. Example: x2 - 7x +10

1. lim x2 x + 2

4

4.

2

( ) x3 - 2x -1 3

2. lim x2 5x2 - 4

-4

-2

-2

2

4

x2 -1 3. lim

x1 x -1

-4

a. Find f (3) . b. Find lim f (x) .

x3

Honors Precalculus

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II. Limits that fail to exist Common types of behavior associated with limits that fail to exist.

A. f (x) approaches a different number from the right side of c than it approaches from

x-3 the left side. ? lim

x3 x - 3

B.

f (x) increases or decreases without bound as x approaches c. ?

1

lim

x1

(

x

-

1)2

C.

f (x) oscillates between two fixed values as x approaches c. ?

lim

x1

sin

x

1 -1

II. Strategy for finding limits. A. Learn to recognize which limits can be evaluated by direct substitution. B. If the limit f(x) as x approaches c CANNOT be evaluated by direct substitution, C. Try to rewrite the function in another form g that agrees with f for all x except x = c. D. Try trig substitutions for special trig limits

E. Examples: 1. lim 4x2 + 3x

x0

x2 + x + 2 2. lim

x1 x + 1

3. lim x cos x x

4. lim x2 -16 x1 3(x - 4)

x3 -8 5. lim

x2 x - 2

(x - 2)2 - 4

6. lim

x0

x

HW. p. 946 ? 1, 2, 4, 12-14 all, 15-37 odds, 43

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Section 15.2A ? The Tangent Line Problem and Derivatives Objectives: 1. To estimate the slope of a curve at an indicated point. 2. To use the definition of the derivative to find the slope of a tangent line to a point on a curve and determine the

equation of the tangent line. 3. To use the rules of differentiation to calculate derivatives

I.

Tangent Line Problem

A. How do we find the equation of the tangent line?

B. Essentially, the problem of finding tangent line at a point P boils

down to the problem of finding the slope of the tangent line at point P

C. Can we approximate the slope using a secant line which contains the points P and Q? How will this help?

D. Slope of Tangent Line 1. Gives an equation:

y lim = lim

x x 0

h0

2. Gives just a value:

lim

xc

E. Examples:

1. Find an equation that will find all the tangent line slopes to the graph f ( x) = x2 + 1

2. Find the slope of the tangent line to the graph of f ( x) = x2 - 2x + 1 at the point (2,1).

II.

Derivative of a Function (Rate of Change at a given moment in time)

A. f (x) = B. Symbols:

The derivative is a function that gives the slope of the tangent line to the graph of f at the point (x, f (x)), provided that the graph has a tangent line at that point.

C. Example: Using the definition of derivative find f (x) if f (x) = 2x2 - 3x + 4

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III. Basic Differential A. Rules of Differentiation d 1. The Constant Rule: (c) = dx 2. The Power Rule: d (xn ) = dx

3. The Constant Multiple Rule: d kxn =

dx

4. Sum and Difference Rule: If f ( x) = g ( x) ? h ( x) , then f ( x) =

B. Examples 1. y = 5

2. f (x) = 3x5 - 2x2 + 1

3. f (x) = (3x - 2)2

4. Find the slope of the graph of f (x) = 2x2 - x -1 when x = -1

IV. Velocity and Acceleration

s

A. Instantaneous Velocity is lim = lim

t t 0

h0

of position:

therefore velocity is the _______________

B. Similarly we can say, acceleration is the ______________ of velocity:

C. Example: At time t = 0 , a diver jumps from a diving board that is 32-ft above the water. The position

of the diver is given by s(t) = -16t 2 + 16t + 32 where s is measured in feet and t is measured in seconds. 1. When does the diver hit the water?

2. What is the divers velocity at impact?

3. During what interval is the diver moving upward?

4. When does the diver reach a maximum height? What is the maximum height?

5. What is the divers acceleration after 1 second?

Homework: p.957 ? 4-9 all, 13, 21-33 odds, 46 a-c, 47, 49, 51, 52

Honors Precalculus

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Section 15.2B ? Graphs and Critical Points of a Polynomial Objectives: 1. To find the critical points and determine if it is a maximum or minimum. 2. To find the points of inflection/

I. Critical Points A. Definition: Critical points occur when the ___________ of the tangent line equals ________. B. Relative Extrema 1. Relative Maximum: a) Slope goes from ____________ to ______________. b) Curve changes from _____________ to ______________. 2. Relative Minimum a) Slope goes from _____________ to ________________. b) Curve changes from _____________ to _______________. C. Example: y = 2x3 - 24x +12

II. Points of Inflection A. Definition: Points of inflections occur when the graph changes its _____________. This happens when the __________ derivative (__________ of the ____________) equals ________. B. Example: y = 2x3 - 24x +12

General Examples 1. The point at (2,15) is a critical point of the graph f (x) = -4x2 +16x -1. Determine if this is

a maximum or a minimum.

2. Find the critical points and points of inflection of the graph f (x) = -2x3 + 3x2 - 5 . Determine if the critical point(s) is a local minimum or local maximum.

3. One hour after x milligrams of a particular drug are given to a person, the rise in body

temperature T ( x) , in degrees Fahrenheit, is given by T ( x) = x - x2 . Determine the number

9 of milligrams that will cause the greatest change in temperature. How many degrees can the patient expect their body temperature to change if given that dose?

Homework: p.176 ? 1-5 all, 8-12 all, 25-33 odds, 39

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Section 15.2C ? Antiderivatives and Indefinite Integration Objectives: 1. To find general antiderivatives of un and linear combinations of these functions. 2. To apply basic integration rules for indefinite integrals

I. Antiderivatives A. Notes: 1. Finding the antiderivative of a function is the _____________ of finding the derivative.

2. For a function f ( x) , the antiderivative is often denoted by F ( x). 3. The relationship between the two functions is F( x) = f ( x) .

B. Example: What is an antiderivative of the following functions?

1. f ( x) = x then F ( x) = 2. f ( x) = 3x2 then F ( x) = 3. f ( x) = 7 then F ( x) =

II. Basic Antiderivative Rules

A. General Power Rule: If f ( x) = kxn , then F ( x) =

B. Examples

1. f ( x) = 32x3

2. f ( x) = 35x6 +12x2 - 6x +12

3. f ( x) = x2 ( x2 + x + 3)

III. Initial Conditions and Particular Solutions Examples: 1. Find the equation of the curve whose slope at a point (x, y) is 3x , if the curve is required to pass through the point (1, -1) .

2. The velocity of a particle is given by v (t ) = 3t2 - 24t + 36 , where t is the time

in seconds. Given the particle's position at 2 seconds is 50, find the equation of for the position of the particle.

Homework: p. 957 ? 2, 10-12 all, 35-45 all, 46 d-e

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Section 15.3 ? Calculus and Area Objectives: 1. To approximate the area under a curve using Riemann Sums of a nonnegative continuous

function.

I. Riemann Sums A. Definite Integral ? If f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f, the x-axis, and the vertical line x = a

and x = b is given by Area = b f (x)dx a

B. Method:

1.

a

b

II. Four Types A. Rectangular Method 1. Left Sided Rectangles

2. Middle Rectangles

3. Right Sided Rectangles

B. Trapezoid Method

III. Theorem:

A. Area = b f (x)dx = lim LRAM = lim MRAM = lim RRAM = limTAM where n is the

a

n

n

n

n

number of rectangles or trapezoids.

Homework: Worksheet

Honors Precalculus

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Precalculus Worksheet Section 15.3

Do on graph paper. Graphs must be exact and visibly large. n represents the number of divisions

( ) 1. Estimate 6 x2 +1 dx using the left-rectangular method with n = 3 without the use of your 0 calculator. Represent the sum graphically on a sketch of f (x) = x2 +1 .

( ) 2. Estimate 6 x2 +1 dx using the right-rectangular method sum with n = 3 without the use of 0 your calculator. Represent the sum graphically on a sketch of f (x) = x2 +1 .

( ) 3. Estimate 6 x2 +1 dx using the mid-rectangular method with n = 3 without the use of your 0 calculator. Represent the sum graphically on a sketch of f (x) = x2 +1 .

( ) 4. Estimate 6 x2 +1 dx using the trapezoid method with n = 3 without the use of your 0 calculator. Represent the sum graphically on a sketch of f (x) = x2 +1 .

Write the following as integrals and then approximate the sums using one of the four methods for the indicated divisions. Each method must be used once. Do on graph paper. Graphs must be exact and visibly large 5. y = x for x = 0 to x = 1 with 4 divisions.

6. y = 1 for x = 4 to x = 6 with 4 divisions. x-2

7. y = 1- x2 for x = 0 to x = 1 with 5 divisions.

8. y = tan x for x = 0 to x = with 6 divisions. 4

9. Find the area of the regions bound by y = 4 - x2 and the x-axis with 8 divisions.

10. y = 1 for x = 0 to x = 2 with 4 divisions. 1+ x3

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