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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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(A) Features You Should Notice About A Graph

(A) Features You Should Notice About A Graph

(1) Basic Curves

(A) Features You Should Notice About A Graph

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

(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)

(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)

(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

(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)

(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

(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

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)

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)

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)

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)

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)

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

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

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

(2) Odd & Even Functions

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

(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)

(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)

(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

(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)

(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

(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?

(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

(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

(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

(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

(5) Asymptotes

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

(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

(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)

(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

(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.

(B) Using 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.

(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.

(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.

(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.

(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.

(3) Minimum/Maximum Turning Points

(3) Minimum/Maximum Turning Points

,0 and 0 When a) 2

2

dx

yddxdy the point is called a minimum turning point

(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

(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:

(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

(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

(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:

(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:

(5) Increasing/Decreasing Curves

(5) Increasing/Decreasing Curves

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

thus increasing

(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

(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

(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

(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

(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

x

y y=x

(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)

x

y y=x

1

1

(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

(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

x

y y=x

1

1

y=-x

(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;

(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

(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

x

y y=x

1

1

y=-x

133 yx

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