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X2 T04 01 curve sketching - basic features/ calculus

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(A) Features You Should Notice About A Graph
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Page 1: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

Page 2: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic Curves

Page 3: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;

Page 4: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;a) Straight lines: xy (both pronumerals are to the power of one)

Page 5: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;a) Straight lines:

b) Parabolas:

xy (both pronumerals are to the power of one)2xy (one pronumeral is to the power of one, the other

the power of two)

Page 6: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;a) Straight lines:

b) Parabolas:

xy (both pronumerals are to the power of one)2xy (one pronumeral is to the power of one, the other

the power of two)NOTE: general parabola is cbxaxy 2

Page 7: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;a) Straight lines:

b) Parabolas:

c) Cubics:

xy (both pronumerals are to the power of one)2xy (one pronumeral is to the power of one, the other

the power of two)NOTE: general parabola is cbxaxy 2

3xy (one pronumeral is to the power of one, the other the power of three)

Page 8: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;a) Straight lines:

b) Parabolas:

c) Cubics:

xy (both pronumerals are to the power of one)2xy (one pronumeral is to the power of one, the other

the power of two)NOTE: general parabola is cbxaxy 2

3xy (one pronumeral is to the power of one, the other the power of three)

NOTE: general cubic is dcxbxaxy 23

Page 9: X2 T04 01 curve sketching - basic features/ calculus

(A) Features You Should Notice About A Graph

(1) Basic CurvesThe following basic curve shapes should be recognisable from the equation;a) Straight lines:

b) Parabolas:

c) Cubics:

xy (both pronumerals are to the power of one)2xy (one pronumeral is to the power of one, the other

the power of two)NOTE: general parabola is cbxaxy 2

3xy (one pronumeral is to the power of one, the other the power of three)

NOTE: general cubic is dcxbxaxy 23

d) Polynomials in general

Page 10: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas: 1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

Page 11: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)

Page 12: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

g) Circles:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)222 ryx (both pronumerals are to the power of two,

coefficients are the same)

Page 13: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

g) Circles:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)222 ryx (both pronumerals are to the power of two,

coefficients are the same)

h) Ellipses: kbyax 22 (both pronumerals are to the power of two, coefficients are NOT the same)

Page 14: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

g) Circles:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)222 ryx (both pronumerals are to the power of two,

coefficients are the same)

h) Ellipses: kbyax 22 (both pronumerals are to the power of two, coefficients are NOT the same)

(NOTE: if signs are different then hyperbola)

Page 15: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

g) Circles:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)222 ryx (both pronumerals are to the power of two,

coefficients are the same)

h) Ellipses: kbyax 22 (both pronumerals are to the power of two, coefficients are NOT the same)

(NOTE: if signs are different then hyperbola)i) Logarithmics: xy alog

Page 16: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

g) Circles:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)222 ryx (both pronumerals are to the power of two,

coefficients are the same)

h) Ellipses: kbyax 22 (both pronumerals are to the power of two, coefficients are NOT the same)

(NOTE: if signs are different then hyperbola)i) Logarithmics: xy alog

j) Trigonometric: xyxyxy tan,cos,sin

Page 17: X2 T04 01 curve sketching - basic features/ calculus

e) Hyperbolas:

f) Exponentials:

g) Circles:

1 OR 1 xy

xy

(one pronomeral is on the bottom of the fraction, the other is not OR pronumerals are multiplied together)

xay (one pronumeral is in the power)222 ryx (both pronumerals are to the power of two,

coefficients are the same)

h) Ellipses: kbyax 22 (both pronumerals are to the power of two, coefficients are NOT the same)

(NOTE: if signs are different then hyperbola)i) Logarithmics: xy alog

j) Trigonometric: xyxyxy tan,cos,sin

k) Inverse Trigonmetric: xyxyxy 111 tan,cos,sin

Page 18: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even Functions

Page 19: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

Page 20: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

Page 21: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)

Page 22: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = x

Page 23: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)

Page 24: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)(4) Dominance

Page 25: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)(4) DominanceAs x gets large, does a particular term dominate?

Page 26: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)(4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates

dominates ,223 e.g. 434 xxxxy

Page 27: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)(4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates

dominates ,223 e.g. 434 xxxxy b) Exponentials: tends to dominate as it increases so rapidlyxe

Page 28: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)(4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates

dominates ,223 e.g. 434 xxxxy b) Exponentials: tends to dominate as it increases so rapidlyxec) In General: look for the term that increases the most rapidly

i.e. which is the steepest

Page 29: X2 T04 01 curve sketching - basic features/ calculus

(2) Odd & Even FunctionsThese curves have symmetry and are thus easier to sketch

xfxf :ODD a) (symmetric about the origin, i.e. 180 degree rotational symmetry)

xfxf :EVEN b) (symmetric about the y axis)(3) Symmetry in the line y = xIf x and y can be interchanged without changing the function, the curve is relected in the line y = x

1 e.g. 33 yx (in other words, the curve is its own inverse)(4) DominanceAs x gets large, does a particular term dominate?a) Polynomials: the leading term dominates

dominates ,223 e.g. 434 xxxxy b) Exponentials: tends to dominate as it increases so rapidlyxec) In General: look for the term that increases the most rapidly

i.e. which is the steepestNOTE: check by substituting large numbers e.g. 1000000

Page 30: X2 T04 01 curve sketching - basic features/ calculus

(5) Asymptotes

Page 31: X2 T04 01 curve sketching - basic features/ calculus

(5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zero

Page 32: X2 T04 01 curve sketching - basic features/ calculus

(5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zero

b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero

Page 33: X2 T04 01 curve sketching - basic features/ calculus

(5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zero

b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero

NOTE: if order of numerator order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check)

Page 34: X2 T04 01 curve sketching - basic features/ calculus

(5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zero

b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero

NOTE: if order of numerator order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check)

(6) The Special Limit

Page 35: X2 T04 01 curve sketching - basic features/ calculus

(5) Asymptotesa) Vertical Asymptotes: the bottom of a fraction cannot equal zero

b) Horizontal/Oblique Asymptotes: Top of a fraction is constant, the fraction cannot equal zero

NOTE: if order of numerator order of denominator, perform a polynomial division. (curves can cross horizontal/oblique asymptotes, good idea to check)

(6) The Special LimitRemember the special limit seen in 2 Unit 1sinlim i.e.

0

xx

x

, it could come in handy when solving harder graphs.

Page 36: X2 T04 01 curve sketching - basic features/ calculus

(B) Using Calculus

Page 37: X2 T04 01 curve sketching - basic features/ calculus

(B) Using CalculusCalculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.

Page 38: X2 T04 01 curve sketching - basic features/ calculus

(B) Using Calculus

(1) Critical Points

Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.

Page 39: X2 T04 01 curve sketching - basic features/ calculus

(B) Using Calculus

(1) Critical Points

dxdy

When is undefined the curve has a vertical tangent, these points are called critical points.

Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.

Page 40: X2 T04 01 curve sketching - basic features/ calculus

(B) Using Calculus

(1) Critical Points

(2) Stationary Points

dxdy

When is undefined the curve has a vertical tangent, these points are called critical points.

Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.

Page 41: X2 T04 01 curve sketching - basic features/ calculus

(B) Using Calculus

(1) Critical Points

(2) Stationary Points

dxdy

When is undefined the curve has a vertical tangent, these points are called critical points.

Calculus is still a tremendous tool that should not be disregarded when curve sketching. However, often it is used as a final tool to determine critical points, stationary points, inflections.

0dxdy

When the curve is said to be stationary, these points may be minimum turning points, maximum turning points or points of inflection.

Page 42: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

Page 43: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

Page 44: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

,0 and 0 When b) 2

2

dx

yddxdy the point is called a maximum turning point

Page 45: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

,0 and 0 When b) 2

2

dx

yddxdy the point is called a maximum turning point

functions

harder for quicker becan changefor of sideeither testingdxdyNOTE:

Page 46: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

,0 and 0 When b) 2

2

dx

yddxdy the point is called a maximum turning point

functions

harder for quicker becan changefor of sideeither testingdxdyNOTE:

(4) Inflection Points

Page 47: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

,0 and 0 When b) 2

2

dx

yddxdy the point is called a maximum turning point

functions

harder for quicker becan changefor of sideeither testingdxdyNOTE:

(4) Inflection Points

,0 and 0 When a) 3

3

2

2

dx

yddx

yd the point is called an inflection point

Page 48: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

,0 and 0 When b) 2

2

dx

yddxdy the point is called a maximum turning point

functions

harder for quicker becan changefor of sideeither testingdxdyNOTE:

(4) Inflection Points

,0 and 0 When a) 3

3

2

2

dx

yddx

yd the point is called an inflection point

functions

harder for quicker becan changefor of sideeither testing 2

2

dxydNOTE:

Page 49: X2 T04 01 curve sketching - basic features/ calculus

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

,0 and 0 When b) 2

2

dx

yddxdy the point is called a maximum turning point

functions

harder for quicker becan changefor of sideeither testingdxdyNOTE:

(4) Inflection Points

,0 and 0 When a) 3

3

2

2

dx

yddx

yd the point is called an inflection point

,0 and 0,0 When b) 3

3

2

2

dx

yddx

yddxdy the point is called a horizontal

point of inflection

functions

harder for quicker becan changefor of sideeither testing 2

2

dxydNOTE:

Page 50: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

Page 51: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

Page 52: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing

Page 53: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation

Page 54: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

Page 55: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yx

Page 56: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

Page 57: X2 T04 01 curve sketching - basic features/ calculus

x

y y=x

Page 58: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

• it passes through (1,0) and (0,1)

Page 59: X2 T04 01 curve sketching - basic features/ calculus

x

y y=x

1

1

Page 60: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

• it passes through (1,0) and (0,1)• it is asymptotic to the line y = -x

Page 61: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

• it passes through (1,0) and (0,1)• it is asymptotic to the line y = -x

xyxyxy

33

33

i.e.1

Page 62: X2 T04 01 curve sketching - basic features/ calculus

x

y y=x

1

1

y=-x

Page 63: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

• it passes through (1,0) and (0,1)• it is asymptotic to the line y = -x

xyxyxy

33

33

i.e.1

On differentiating implicitly;

Page 64: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

• it passes through (1,0) and (0,1)• it is asymptotic to the line y = -x

xyxyxy

33

33

i.e.1

On differentiating implicitly;

2

2

22 033

yx

dxdydxdyyx

Page 65: X2 T04 01 curve sketching - basic features/ calculus

(5) Increasing/Decreasing Curves

,0 When a) dxdy the curve has a positive sloped tangent and is

thus increasing

,0 When b) dxdy the curve has a negative sloped tangent and is

thus decreasing(6) Implicit Differentiation This technique allows you to differentiate complicated functions

1Sketch e.g. 33 yxNote:• the curve has symmetry in y = x

• it passes through (1,0) and (0,1)• it is asymptotic to the line y = -x

xyxyxy

33

33

i.e.1

On differentiating implicitly;

2

2

22 033

yx

dxdydxdyyx

This means that for all x

Except at (1,0) : critical point &

(0,1): horizontal point of inflection

0dxdy

Page 66: X2 T04 01 curve sketching - basic features/ calculus

x

y y=x

1

1

y=-x

133 yx


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