Differentiation Flashcards
What is the gradient of the tangent equal to
The gradient of the curve at a specified point
Find the gradient and equation of the tangent to y = 3x^2 at x = 1
y = 6x - 3
Method for finding equation of a tangent
- 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
A parabola has the equation y = x^2 - 4x + 1. Calculate the gradient and equation of the tangent at x = 3.
y = 2x - 8
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
y = 32x - 47
Find the interval for which the function y = 3x^2 + 2x - 5 is increasing
function is increasing when x > -1/3
Finding interval for when a function is increasing method
- find derivative
- set derivative > 0
- solve for x
- state interval
Finding interval when function is decreasing method
- find derivative
- set derivative < 0
- solve for x
- state interval
If a graph slopes down from left to right, the function is ________
Decreasing
If a graph slopes up from left to right, the function is ________
Increasing
The derivative of a function is equal to what
The gradient
If the derivative > 0, the curve is what
Increasing as m is positive
If the curve is increasing, what is the value of the derivative
> 0
If the derivative < 0, the curve is what
The curve is decreasing as m is negative
Value of derivative when curve is decreasing
< 0
Method for determining if function is increasing or decreasing
- find derivative
- sub x into derivative
- stare if function is increasing or decreasing
Is y = x^2 - 5x increasing or decreasing when x=4?
Increasing at x = 4 since dy/dx > 0.
(= 3)
Stationary point
Graph is neither increasing or decreasing
Gradient at stationary points
0
Nature of stationary point
- Maximum turning point
- Minimum turning point
- Point of inflection
- Maximum turning point
- Minimum turning point
- Point of inflection
Nature of turning point Minimum
Method for finding stationary point
- differentiate and set equal to 0
- solve to find x
- nature table
- sub x into original equation to find y
Minimum stationary point at (3, -11)
Maximum stationary point at (-1, 4)
Minimum stationary point at (1, -4)
B) increasing
Minimum turning point (-1, -2)
POI (0, 0)
Maximum turning points (1, 2)
The gradient of the tangent is equal to what
The gradient of the curve at a specific point
The gradient of the curve at a specific point is equal to what
The gradient of the tangent
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
Find the gradient and equation of the tangent to y = 3x^2 at x = 1
y = 6x - 3
point = (1, 3)
m = 6
y = 2x - 8
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
Stationary point
Graph that is neither increasing or decreasing
Graph that is neither increasing or decreasing
Stationary point
Gradient at stationary points
0
Nature of stationary point can be…
- Maximum turning point
- Minimum turning point
- Point of inflection
- Maximum turning point
- Minimum turning point
- Point of inflection
Nature of stationary point
Method for finding stationary point
- Differentiate and set equation = 0
- Solve to find x
- Nature table
- Sub x into original equation to find y
- Differentiate and set equation = 0
- Solve to find x
- Nature table
- Sub x into original equation to find y
Method for finding stationary point
Minimum stationary point at (3, -11)
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
Minimum TP (3, -18)
Maximum value = 32
Minimum value = -18
a) 17 and 189
b) POI (0, 0), minimum TP (2, -64)
c) max value = 189, min value = -64
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
Find the roots of 3x^2 - 12 = 0
2 and -2
What is differentiating finding
The rate of change
What finds the rate of change
Differentiate
y when differentiated…
dy / dx
How to differentiate
Multiply the coefficient by the power and decrease the power by 1
6x^2
2x
2x + 10
27
75
a0 = ?
1
a^-m = ?
1 / a^m
Write with positive indices:
Write with positive indices:
Write in the form ax^n:
Write in the form ax^n:
Write in the form ax^n:
Write in the form ax^n:
Simplify:
Simplify:
x / 5
Simplify:
What to ensure before differentiating
- no x term on the bottom of a fraction
- change square root to fractional indices
How to subtract ‘1’ from a fraction
Top - bottom
3/2 - 1
1/2
6/5 - 1
1/5
4/7 - 1
-3/7
-4/5 - 1
-9/5
-7/3 - 1
-10/3
-1/2 - 1
-3/2
Differentiate 1/x^3
-3x^-4
Differentiate:
Differentiate:
Find f ’(9) for the following:
9/2
Differentiate:
Displacement
- Distance from the origin at time
- A distance with direction
- Units: metres
- Distance from the origin at time
- A distance with direction
- Units: metres
Displacement
Velocity
- Rate of change of displacement with respect to time
- A speed with direction
- Units: m/s
- Rate of change of displacement with respect to time
- A speed with direction
- Units: m/s
Velocity
Acceleration
- The rate of change of velocity with respect to time
- The rate at which an object changes its speed
- Units: m/s^2
- The rate of change of velocity with respect to time
- The rate at which an object changes its speed
- Units: m/s^2
Acceleration
Relationship between displacement, velocity, and acceleration
Displacement
V
Velocity
V
Acceleration
By differentiation
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
A car travelling has a velocity of v = 10 + 6t^2 - t^3
Find the acceleration when t = 3
9 m/s^2
a) velocity = -4 + 3t^2
Acceleration = 6t
b) displacement = -2m
velocity = -1m/s
acceleration = 6 m/s^2
c) 2 seconds
Name, equation and degree of the following graph:
Constant
Y = a
Degree = 0
Name, equation and degree of the following graph:
Linear graph
Y = mx + c
Degree = 1
Name, equation and degree of the following graph:
Name, equation and degree of the following graph:
Name, equation and degree of the following graph:
What happens to the graph when you differentiate a function
It’s degree is reduced by one. This corresponds to a graph
f (x) gradient -> f’(x) graph relationships
+ve -> above x axis
zero -> on x axis
-ve -> below x axis
Express with +ive indices:
Write with +ive indices:
Write with +ive indices:
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)
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)
POI at (0, 0)
Max TP at (2, 16)
y intercept at (0, 0)
x intercept at (0, 0) and (8/3, 0)
