Differentiation Flashcards

1
Q

What is the gradient of the tangent equal to

A

The gradient of the curve at a specified point

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2
Q

Find the gradient and equation of the tangent to y = 3x^2 at x = 1

A

y = 6x - 3

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3
Q

Method for finding equation of a tangent

A
  • differentiate the equation given
  • sub x into differentiated equation to find m
  • find y, state x and y in the form (x, y)
  • find equation of tangent by subbing known data into equation
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4
Q

A parabola has the equation y = x^2 - 4x + 1. Calculate the gradient and equation of the tangent at x = 3.

A

y = 2x - 8

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5
Q

The gradient of a tangent to the curve y = x^4 + 1 is 32.

a) find the point of contact of the tangent
b) find the equation of the tangent

A

y = 32x - 47

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6
Q

Find the interval for which the function y = 3x^2 + 2x - 5 is increasing

A

function is increasing when x > -1/3

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7
Q

Finding interval for when a function is increasing method

A
  • find derivative
  • set derivative > 0
  • solve for x
  • state interval
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8
Q

Finding interval when function is decreasing method

A
  • find derivative
  • set derivative < 0
  • solve for x
  • state interval
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9
Q

If a graph slopes down from left to right, the function is ________

A

Decreasing

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10
Q

If a graph slopes up from left to right, the function is ________

A

Increasing

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11
Q

The derivative of a function is equal to what

A

The gradient

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12
Q

If the derivative > 0, the curve is what

A

Increasing as m is positive

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13
Q

If the curve is increasing, what is the value of the derivative

A

> 0

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14
Q

If the derivative < 0, the curve is what

A

The curve is decreasing as m is negative

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15
Q

Value of derivative when curve is decreasing

A

< 0

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16
Q

Method for determining if function is increasing or decreasing

A
  • find derivative
  • sub x into derivative
  • stare if function is increasing or decreasing
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17
Q

Is y = x^2 - 5x increasing or decreasing when x=4?

A

Increasing at x = 4 since dy/dx > 0.

(= 3)

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18
Q
A
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19
Q
A
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20
Q

Stationary point

A

Graph is neither increasing or decreasing

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21
Q

Gradient at stationary points

A

0

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22
Q

Nature of stationary point

A
  • Maximum turning point
  • Minimum turning point
  • Point of inflection
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23
Q
  • Maximum turning point
  • Minimum turning point
  • Point of inflection
A

Nature of turning point Minimum

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24
Q

Method for finding stationary point

A
  • differentiate and set equal to 0
  • solve to find x
  • nature table
  • sub x into original equation to find y
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25
Minimum stationary point at (3, -11)
26
Maximum stationary point at (-1, 4) Minimum stationary point at (1, -4) B) increasing
27
Minimum turning point (-1, -2) POI (0, 0) Maximum turning points (1, 2)
28
The gradient of the tangent is equal to what
The gradient of the curve at a specific point
29
The gradient of the curve at a specific point is equal to what
The gradient of the tangent
30
Method for finding the equation of a tangent when given equation and x?
- differentiate equation - sub x into derivative to find m - find y, state x and y as (x, y) - find equation by subbing point and gradient
31
Find the gradient and equation of the tangent to y = 3x^2 at x = 1
y = 6x - 3 point = (1, 3) m = 6
32
y = 2x - 8
33
The gradient of a tangent to the curve y = x^4 + 1 is 32. a) find point of contact b) find equation
(2, 17) y = 32x - 47
34
Stationary point
Graph that is neither increasing or decreasing
35
Graph that is neither increasing or decreasing
Stationary point
36
Gradient at stationary points
0
37
Nature of stationary point can be…
- Maximum turning point - Minimum turning point - Point of inflection
38
- Maximum turning point - Minimum turning point - Point of inflection
Nature of stationary point
39
Method for finding stationary point
- Differentiate and set equation = 0 - Solve to find x - Nature table - Sub x into original equation to find y
40
- Differentiate and set equation = 0 - Solve to find x - Nature table - Sub x into original equation to find y
Method for finding stationary point
41
Minimum stationary point at (3, -11)
42
When do the max and min values of a function occur in a closed interval?
The maximum or minimum turning points or at the ends of the interval
43
Minimum TP (3, -18) Maximum value = 32 Minimum value = -18
44
a) 17 and 189 b) POI (0, 0), minimum TP (2, -64) c) max value = 189, min value = -64
45
method for finding max and min values of a function in a closed interval
- sub max value into function - sub min value into function - find stationary points using derivative - state largest and smallest values
46
Find the roots of 3x^2 - 12 = 0
2 and -2
47
What is differentiating finding
The rate of change
48
What finds the rate of change
Differentiate
49
y when differentiated…
dy / dx
50
How to differentiate
Multiply the coefficient by the power and decrease the power by 1
51
6x^2
52
2x
53
54
55
56
2x + 10
57
27
58
75
59
a0 = ?
1
60
a^-m = ?
1 / a^m
61
62
63
64
65
Write with positive indices:
66
Write with positive indices:
67
Write in the form ax^n:
68
Write in the form ax^n:
69
Write in the form ax^n:
70
Write in the form ax^n:
71
Simplify:
72
Simplify:
x / 5
73
Simplify:
74
What to ensure before differentiating
- no x term on the bottom of a fraction - change square root to fractional indices
75
How to subtract ‘1’ from a fraction
Top - bottom
76
3/2 - 1
1/2
77
6/5 - 1
1/5
78
4/7 - 1
-3/7
79
-4/5 - 1
-9/5
80
-7/3 - 1
-10/3
81
-1/2 - 1
-3/2
82
Differentiate 1/x^3
-3x^-4
83
Differentiate:
84
Differentiate:
85
Find f ’(9) for the following:
9/2
86
Differentiate:
87
Displacement
- Distance from the origin at time - A distance with direction - Units: metres
88
- Distance from the origin at time - A distance with direction - Units: metres
Displacement
89
Velocity
- Rate of change of displacement with respect to time - A speed with direction - Units: m/s
90
- Rate of change of displacement with respect to time - A speed with direction - Units: m/s
Velocity
91
Acceleration
- The rate of change of velocity with respect to time - The rate at which an object changes its speed - Units: m/s^2
92
- The rate of change of velocity with respect to time - The rate at which an object changes its speed - Units: m/s^2
Acceleration
93
Relationship between displacement, velocity, and acceleration
Displacement V Velocity V Acceleration By differentiation
94
A car is travelling along a straight road. The distance, x metres, travelled in t seconds is x = 10t - 5t^2 Find the velocity when t = 0.5
5 m/s
95
A car travelling has a velocity of v = 10 + 6t^2 - t^3 Find the acceleration when t = 3
9 m/s^2
96
a) velocity = -4 + 3t^2 Acceleration = 6t b) displacement = -2m velocity = -1m/s acceleration = 6 m/s^2 c) 2 seconds
97
Name, equation and degree of the following graph:
Constant Y = a Degree = 0
98
Name, equation and degree of the following graph:
Linear graph Y = mx + c Degree = 1
99
Name, equation and degree of the following graph:
100
Name, equation and degree of the following graph:
101
Name, equation and degree of the following graph:
102
What happens to the graph when you differentiate a function
It’s degree is reduced by one. This corresponds to a graph
103
f (x) gradient -> f’(x) graph relationships
+ve -> above x axis zero -> on x axis -ve -> below x axis
104
105
106
107
Express with +ive indices:
108
Write with +ive indices:
109
Write with +ive indices:
110
To sketch a curve, what do we need to know?
- it’s stationary points and their nature - where it crosses the x axis (set y=0) - where it crosses the y axis (set x = 0)
111
Min TP (-1, -2) Max TP (1, 2) Graph passes y axis at (0, 0) Graph crosses x axis at (/3, 0) and (-/3, 0)
112
POI at (0, 0) Max TP at (2, 16) y intercept at (0, 0) x intercept at (0, 0) and (8/3, 0)
113
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115