MST124 BOOK B UNIT 6

Gradient of a straight line
The gradient of the straight line through the points (x₁,y₁) and (x₂,y₂), where x₁ does not equal x₂, is given by
gradient = rise/run = (y₂-y₁)/(x₂-x₁)

- the gradient of a graph at a particular point is the gradient of the tangent to the graph at that point 

- consider the graph of y = magnitude of x
- there is no tangent to the graph at the origin, so the graph doesn't have a gradient at this point
- in general, if a graph has a 'sharp corner' at a point, then it has no gradient at that point 
- also, if a graph has a 'break' (known as a discontinuity) at a particular point, then the graph has no tangent, and hence no gradient, at this point
- if a graph has a vertical tangent at a point, then the graph has no gradient at that point as this would be undefinable 

- consider a function f and a particular input value x
- the point on the graph of f that corresponds to the input value x is (x,f(x))
- if the graph of f has a gradient at the point (x,f(x)), then we say that f is differentiable at x
- for example, the function f(x) = x² is differentiable at every value of x
- if the graph of f doesn't have a gradient at the point (x, f(x)), then f isn't differentiable at x
- similarly, if a function of f isn't even defined at a particular input value x, then it's not differentiable at x 

- the gradient of the graph of a function f varies depending on which value of x you're considering
- its convenient to think of these gradients as defining a new function, related to f
- the rule of this new function would be:
if the input value is x, then the output value is the gradient of the graph of f at the point (x,f(x)) 

- this new function is called the derivative (or derived function) of the function f
- it is denoted by f'
- the domain of the derivative consists of all the values at which f is differentiavle
- the process of finding the derivative of a given function is called differentiation 

- the value of the derivative of a function f at a particular input value x is called the derivative of f at x 

(f(x+h) - f(x))/h is known as the difference quotient for the function f at the value x 
- as the second point (x + h, f(x + h)) gets closer and closer to the first point (x,f(x)) (h gets closer to 0) the value of the difference quotient gets closer and closer to the gradient of the graph at the point (x, f(x))
- closer to f'(x)

Differentiation from first principles
For any function f, the derivative f' of f is given by the equation 
f'(x) = lim_{h to 0}(f(x+h) - f(x))/h 
for each value of x in the domain of f for which this limit exists  

- the notation in which the derivative of a function f is denoted by f' is called Lagrange notation or prime notation 
- there is another notation called Leibniz notation

- consider the equation y = x², which expresses a relationship between the variable x and y
- the formula for the gradient of the graph of the equation y = x² is:
gradient = 2x
In Leibniz notation, this equation is written as
dy/dx = 2x
(the derivative of y with respect to x) 

- any function whose domain includes an endpoint isn't differentiable at this endpoint
- consider the function f(x) = x^3/2
- the point on its graph that corresponds to the endpoint 0 of its domain is the origin
- you can trace your pen tip along the graph to the origin from the right, but you can't do the same from the left, so the graph doesn't have a tangent at the origin
- so, it doesn't have a gradient, and therefore is not differentiable at x = 0 

- however, the graph of f(x) = x^3/2 does have a 'tangent on the right' at the origin
- you can trace your pen tip along the graph towards the origin from the right, and continue moving it in the direction in which its been moving when it reaches the origin

(cont.)
- the graph has a 'gradient on the right at the origin, namely 0, the gradient of the x-axis 
- so, this graph is right-differentiable at x = 0, and its right derivative at 0 is 0 
- similarly, a function can be left-differentiable at a particular x-value, and it will then have a left derivative at that x-value 

- the left or right derivative of a function at a particular x-value can be found using differentiation from first principles in the usual way
- the only difference is that instead of the increase in the x-coordinates, h, taking either +ve or -ve values as it gets closer and closer to 0, for a right-sided derivative h takes just +ve values
- for a left-sided derivative h takes just -ve values 

- saying that a function is differentiable at a particular x-value is the same as saying that it has both a left and a right derivative at that x-value 
- the left and right derivatives are equal 

- if f is a function whose domain includes one or more endpoints, then we adjust the definition of its derivative f' slightly to allow for these

- must include in the domain of f' the values of x at which f is differentiable, and the values of x that are the endpoints of the domain of f and at which f is left- or right-differentiable
- the value of f' at each of these endpoints is the appropriate left or right derivative 

DERIVATIVES
The derivative (or derived function) of a function f is the function f' such that
f'(x) = gradient of the graph of f at the point (x, f(x))
The domain of f' consists of the values in the domain of f at which f is differentiable
- that is, the x-values that give points at which the gradient exists
If y = f(x), then f'(x) is also denoted by dy/dx
The derivative f' is given by the equation
f'(x) = lim_{h to 0}(f(x+h) - f(x))/h
The procedure of using this equation to find a formula for the derivative f' is called differentiation from first principles 

Derivative of a power function
For any number n,
d/dx(xⁿ) = nx^n-1 

Two index laws
a^-n = 1/aⁿ
a^1/n = ⁿsqrt(a) 

f(x) = x has derivative
f'(x) = 1*x⁰; that is f'(x) = 1

f(x) = 1 has the simple derivative of f'(x) = 0 

Constant multiple rule (Lagrange notation)
If the function k is given by k(x) = af(x), where f is a function and a is a constant term, then
k'(x) = af'(x)
for all values of x at which f is differentiable 

Constant multiple rule (Leibniz notation)
if y = au, where u is a function of x and a is a constant, then dy/dx = a(du/dx) 
for all values of x at which u is differentiable 

Derivative of a constant function
If a is a constant, then
d/dx(a) = 0

Sum rule (Lagrange notation)
If k(x) = f(x) + g(x), where f and g are functions, then
k'(x) = f'(x) + g'(x),
for all values of x at which both f and g are differentiable 

- it follows from the sum rule, together with the constant multiple rule, that if k(x) = f(x) - g(x), then
k'(x) = f'(x) - g'(x)
for all valyes of x at which both f and g are differentiable 

Sum rule (Leibniz notation)
If y = u + v, where u and v are functions of x, then
dy/dx = du/dx + dv/dx 
for all values of x at which both u and v are differentiable 

Polynomial function:
f(x) = a sum of terms, each of the form axⁿ, where a is a number and n is a non-negative integer 

- every polynomial function (with domain R) is differentiable at every value of x 

Relationship between displacement and velocity
- suppose that an object is moving along a straight line. If t is the time that has elapsed since some chosen point in time, s is the displacement of the object from some chosen reference point, and v is the velocity of the object, then 
v = ds/dt 

Functions increasing or decreasing on an interval
A function f is increasing on the interval I if for all values x₁ and x₂ in I such that x₁<x₂,
f(x₁) < f(x₂)
A function f is decreasing on the interval I if for all values x₁ and x₂ in I such that x₁<x₂,
f(x₁) > f(x₂)
The interval I must be part of the domain of f 

Increasing/decreasing criterion
If f'(x) is +ve for all x in an interval I, then f is increasing on I
If f'(x) is -ve for all x in an interval I, then f is decreasing on I 

- a point at which the gradient of a graph is zero is called a stationary point
- a point is a local maximum if the value taken by the function at that point is larger than at any other point nearby 
- similarly, a point is a local minimum if the value taken by the function at that point is smaller than at any other point nearby 

- a point where a function has a local maximum or local minimum is called a turning point
- if you imagine tracing your pen tip along the graph of the function from left to right, then at a local maximum of minimum it stops going up or down, and turns to go the other way 

- a local maximum of a graph is increasing on the left and decreasing on the right
- at a local minimum, it is decreasing on the left and increasing on the right 

- a stationary point at which it is increasing on the left and right, or decreasing on the left and right is known as a horizontal point of inflection
- the tangent to a curve at a horizontal point of inflection crosses the curve at that point 

- some stationary points are neither turning points nor horizontal points of inflection
- e.g., every point on the graph of the equation y = 1 (or any horizontal line) is a stationary point that is neither a turning point nor a horizontal point of inflection 

Strategy:
To find the stationary points of a function f
Solve the equation f'(x) = 0

- f is increasing on the left interval and decreasing on the right interval if the stationary point is a local minimum
- f is decreasing on the left interval and increasing on the right interval if the stationary point is a local minimum
- f is increasing on both intervals or decreasing on both intervals if the stationary point is a horizontal point of inflection 

First derivative test (for determining the nature of a stationary point of a function f)
- if there are open intervals immediately to the left and right of a stationary point such that:
f'(x) is +ve on the left interval and negative on the right interval, then the stationary point is a local maximum
f'(x) is -ve on the left interval and +ve on the right interval , then the stationary point is a local minimum
f'(x) is +ve on both intervals or -ve on both intervals, then the stationary point is a horizontal point of inflection 

- to use the first derivative test, find f'(x), factories it, work out stationary points, use table of signs

Strategy
To apply the first derivative test by choosing sample points
1 - choose two points (two x-values) fairly close to the stationary point, one on each side
2 - check that the function is differentiable at all points between the chosen points and the stationary point and that there are no other stationary points between the chosen points and the stationary point 
3 - Find the value of the derivative of the function at the two chosen points
- if the derivative is +ve at the left chosen and -ve at the right chosen point, then the stationary point is a local maximum
- if the derivative is -ve at the left chosen point and +ve at the right chosen point, then the stationary point is a local minimum
- if the derivative is +ve at both chosen points or -ve at both chosen points, then the stationary point is a horizontal point of inflection 

Properties of graphs of cubic functions
The graph of every cubic function has the following properties
- there are two, one, or no stationary points
- apart from at any stationary points and in the interval between them if there are two, the graph slopes up from left to right if the coefficient of x³ in the rule of the function is +ve and, and slopes down from left to right if it is -ve
- if there are two stationary points, one is a local maximum and one is a local minimum, and the graph slopes the other way in the interval between them
- if there is one stationary point, then it is a horizontal point of inflection
- there are three, two, or one x-intercepts
- there is only one y-intercept
- the graph tends to + or - infinity for large +ve and large -ve values of x 

Strategy
To find the greatest or least value of a function on an interval of the form [a,b]
(this is valid when the function is continuous on the interval and differentiable at all values in the interval except possibly the endpoints)
- find the stationary points of the function
- find the values of the function at any stationary points inside the interval, and at the endpoints of the interval
- find the greatest or least of the function values found 

- the derivative of a function is a function
- this derivative can be differentiated to obtain a second derivative of the original function 
- denoted by f'' (Lagrange notation) or d²y/dx² 
- the domain of the second derivative of a function consists of all the values of x at which its first derivative is differentiable 
- the original function is twice-differentiable at such values of x

- once a function has been differentiated twice, it can be differentiated three times, then four times, and so on 
- third derivate, f''' or f⁽³⁾, d³y/dx³
- fourth derivative, f'''' or f⁽⁴⁾, d⁴y,dx⁴ 
- the domain of the third derivative of a function consists of all the values of x at which the second derivative is diffferentiable

- a function can be differentiable infinitely many times at a value of 

- the second derivative of a function tells you the gradients of the graph of the first derivative 
- it also gives you information about the shape of the graph of the original function
- suppose that there is an interval on which the second derivative of a particular function is +ve
- this means the first derivative is increasing on that interval, so the gradient of the graph of the original function is increasing on that interval 

- a graph is concave up on an interval if the tangents to the graph on that interval lie below the graph (cross-section of a bowl that is the right way up)

- if there is an interval on which the second derivative of a particular function is -ve, this means the first derivative is decreasing on that interval
- so the gradient of the graph of the original function is to decreasing on that interval

- a graph is concave down on an interval if the tangents to the graph on that interval live above the graph (cross-section of an upside-down bowl) 

- a point where a graph changes from concave up to concave down or vice versa is called a point of inflection 
- if the gradient of a graph at a point of inflection is zero, then the point is a horizontal point of inflection
- otherwise, the point is a slant point of inflection

Concave up/concave down criterion
- if f''(x) is +ve for all x in an interval I, then f is concave up on I
- if f''(x) is -ve for all x in an interval I, then f is concave down on I  

- a graph of a function is concave down at and around a local maximum, and concave up at and around a local minimum 

Second derivative test (for determining the nature of a stationary point)
If, at a stationary point of a function, the value of the second derivative of the function is:
- -ve, then the stationary point is a local maximum
- +ve, then the stationary point is a local minimum 

- if the value of the second derivative of a function at a stationary point is 0, rather than either +ve or -ve, then you can't use the second derivative test to determine the nature of a stationary point
- the stationary point might be a local maximum, a local minimum a horizontal point of inflection, or none of these 

Relationships between displacement, velocity and acceleration
Suppose that an object is moving in a straight line. If t is the time that has elapsed since some chosen point in time, and s, v and a are the displacement, velocity and acceleration of the object:
v = ds/dt, a = dv/t and a =d²s/dt²