Course Notes- Calculus 1

8/12/2019 Course Notes- Calculus 1

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Calculus 1 Name:__________________________________Lesson- Limits and Vertical Asymptotes

Date:___________________________________

Objectives: • determine the limit as the graph approaches a vertical asymptote

Limits:

(1) Graph and find the following limits:

(a)

(b)

(c)


(a)

(b)

(c)

x

y

x

y



19


(a)

(b)

(c)


(a)

(b)

(c)


(a)

(b)

(c)

x

x

y

x

y



20

Calculus 1 Name:__________________________________Lesson- Determining limits graphically

Date:___________________________________

Objectives: • determine the limit of a function graphically


(a)

(b)

(c)


(a)

(b)

(c)

(d)

(e)

(f)

x

y

x

y



21

Given the value of c, use the graph of f to find each of the following values:

(a) (b) (c) (d)

(1) (2)

(4)(3)



22

Calculus 1 Name:__________________________________Lesson - Limits and horizontal asymptotes

Date:___________________________________

Objectives: • determine the limit as the graph approaches a horizontal asymptote

Limits where x approaches infinity: Limits where x approaches a constant

Find each of the following limits:

(1) (2)

(3) (4)

(5) (6)



23

Find the limit for each of the following algebraically:

(1) (2)

(3) (4)

(5) (6)

(7) (8)



24

Graphs, Functions, and Limits & Continuity

Definitions, Properties & Formulas

Linear Equation equation of a straight line

Slope

the slope, m, of the line through (x 1, y1) and (x 2, y2) is given by the following

equation, if x 1 ! x2:

Types of Slope

y-intercept where the graph crosses the y-axis

x-intercept where the graph crosses the x-axis

Slope-InterceptForm

y = mx + b

where m represents the slope and b represents the y-intercept of the linearequation

Standard FormAx + By = C

where A, B, and C are constants and A " 0 (positive, whole number)

Point-SlopeForm

y – y 1 = m(x – x 1)

where m represents the slope and (x 1, y1) are the coordinates of a point on the lineof the linear equation

Parallel Lines

Two nonvertical lines in a plane are parallel if and only if their slopes are equal and they have no points in common. (Two vertical lines are always parallel.)

ex) y = 2x + 3 and y = 2x – 4 ! equal slopesm = 2 m = 2 // lines

PerpendicularLines

Two nonvertical lines in a plane are perpendicular if and only if their slopes arenegative reciprocals . (A horizontal and a vertical line are always perpendicular.)

ex) and!

neg. recip. slopes

m = m = # lines

Relation a set of ordered pairs (x, y)

Domain the set of all x-values of the ordered pairs

Range the set of all y-values of the ordered pairs

NegativePositive Zerohorizontal line:

y = b

Undefinedvertical line: x

= a





26

Increasing,Decreasing, and

ConstantFunctions

Let I be an interval in the domain of a function f. Then:

f is increasing on I iff(b) > f(a) whenever b

> a in I

f is decreasing on I iff(b) < f(a) whenever b

> a in I

f is constant on I if f(b)= f(a) for all a and

b in I

increasing on (- $ ,$ ) decreasing on (- $ ,$ ) constant on (- $ ,$ )

Properties ofLimits

If f is the constant function f(x) = k (the function whose outputs have the constantvalue k), then for any value of c, .

If f is the identity function f(x) = x, then for any value of c,.

If is any polynomial function, then limits can be

found by a substitution method: .

If f(x) and g(x) are polynomials in a rational function (substitution method may

work, but not always), then: .

As : if f is a constant function f(x) = k, then: ; and

if f is the reciprocal function , then .

Right-Hand andLeft-Hand Limits

A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal:

Vertical andHorizontal

Asymptotes

A line y = b is a horizontal asymptote of the graph of a function y = f(x) if either:or

A line x = a is a vertical asymptote of the graph of a function y = f(x) if either:or

Oblique Asymptotes Use Long division to determine the quotient (drop any remainder)



27

Calculus 1 Name:______________________________Review- Units P & 1 Test

Date:_______________________________

SHOW ALL W ORK !

(1) Which of the following equations define functions? Explain your reasoning.

(a) y = x + 6

(b) y2

= x + 1(c) y3 = x + 4

(d) y = x – 2

(e) y3 = x – 3

(f) y2 = x – 5

(2) Which of the following functions are one-to-one? Explain your reasoning.

(a) f(x) = x 5

(b) g(x) = 2x + 7

(c) h(x) = x 2 – 4

(3) Write the linear equation in standard form Ax + By = C that passes through the points (3, 5) and (4, -3).

(4) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3) and is parallel to the line 2x + 2y = 5.



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(5) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3) and is perpendicular to the line 3x – 3y = 4.

(6) Write the linear equation in standard form Ax + By = C that is a horizontal line and passes through the point (-9, 2).

__________________________________________________________________________________________(7) Find the point(s) of intersection between: &

(8) Given f(x) = 2x 2 – x + 3 find f(k – 2)

(9) Given f(x) = 4x – 7 and g(x) = 2x – x 2, evaluate f(2) + g(-1)



29

(10) For the function f(x) = -6x + 5, find and simplify:

(a) (b)

(11) Find the domain of each of the following:

(a) (b)

(12) Algebraically determine the symmetry with respect to the y-axis, x-axis, and origin, if anyexists, for each of the given equations:(a) 2x – 4y = 7 (b) 9x2 – 4y 2 = 36

(13) Algebraically determine if each of the given functions are odd, even, or neither:(a) f(x) = x 2 – 6x(b) f(x) = x 6 + 7



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(14) Perform the four basic operations with the functions f(x) = 4x and g(x) = x 2 + 2

(f + g)(x) =

(f – g)(x) =

(f • g)(x) =

(15) Given f(x) = x 2 – 3 and , find and and give each domain.

(16) For each of the following functions, f(x), find the inverse, f -1(x):(a) f(x) = 5x + 2

(b)

(17) Given the graph on the right, answer the following questions:

(a) write the linear equation in slope-intercept form:

(b) write in slope-intercept form the equation of the parallel linethrough (2, -1) and graph:

(c) write in slope-intercept form the equation of the perpendicularline through the x-intercept and graph:

_______________________________________________________________________________________

x

y



31

Graph each function and use interval notation to answer the following questions:(a) Find the domain

(b) Find the range

(c) Find the intervals over which the function is increasing

(d) Find the intervals over which the function is decreasing

(e) Find the intervals over which the function is constant

(f) State any points of discontinuity

(18) (19)

(20) (21)

x

y

x

y

x

y

x

y



32

(22) (23)

(24) Graph the following piecewise function:

(25) Based on your graph of f(x) from question (7), find the value of each of the following limits:(a) =

(b) =

(c) =

(d) =

(e) =

(f) =

(g) =

(h) =

x

y

x

y

x

y



33

(26) Based on the given graph of f(x), find the value of each of the following:(a) =

(b) =

(c) =

(d) =

(e) =

(f) f(3) =

(g) =

(h) =

Algebraically determine the value each of the following limits or explain why it does not exist:

(27) (28)

(29) (30)

y

x



(31) (32)

(33) (34)

(35) (36)

(37) (38)

(39) (40)

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Course Notes- Calculus 1