and
We sometimes have to sketch graphs of the form
The graph of is a straight line. When, . If, for example if, thenwhich is greater than zero. We can plot the point.
Ifthen . This is less than zero. We are plotting the values ofagainst absolute values of y, so we find the absolute value ofwhich is 1 and plot the point,. At the point whereis zero, on one side of which it would be negative, and on the other side positive, if were were plotting a line and not an absolute value graph, there will be a “corner”.
We can draw up the set of values of x and y. They are given below next to the graph.
In general to sketch any graph of this sort we can start by finding the zero: there will be a corner at that point. On either side of the corner will be a straight line. It is then enough to find one point either side of the corner to draw the graph.
Finding the corner can sometimes be tricky, but you ownly have to find where the modulus part is zero.
For the graphthe corner is whenat the point
For the graphthe corner is when, at the point, shown below.
]]>When adding or subtracting, make a common denominator.
When multiplying fractions, cross cancel first if possible. Multiply the denominators together to make a new denominator. Multiply the numerators together to make a new numerator. Cancel if possible.
When dividing fractions, invert the dividing fraction and multiply.
Example: Add and simplify
Notice first thatfactorises intoThe question becomes
Example: Add and simplify if possible:
No further simplification is possible.
Example: Multiplyby
factorises:
Hence
since theandterms cross cancel.
Example: Divideby
factorises:Hence
Since theterms cross cancel.
]]>wherethe number of strips, is even. Given an integral to estimate, we draw up a table of function values
Example: Using Simpson's Rule with six strips, estimate the value of the integral
x-i |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
f-i |
1 |
2 |
4 |
8 |
16 |
32 |
64 |
Example: Using Simpson's Rule with five strips, estimate the to three decimal places the value of the integralFind the percentage error.
Because the final answer is to be to three decimal places, all figures in the working will be to four decimal places. Only the final answer will be to three decimal places.
x-i |
0 |
2 |
4 |
6 |
8 |
10 |
12 |
f-i |
0 |
1.4142 |
2 |
2.4495 |
2.8284 |
3.1623 |
3.4641 |
to four decimal places so
The true value is
The % error is
]]>For example, ifandthen
It must first be noted that writing a function asoris purely to define the form of the function. The argument of a function is arbitrary. Any expression may be substituted for- a function is defined completely by it's form, so that if we have a functionand we need too find a functionthen we represent all the's in the expression forby's. Ifis also a function of, then we may substitutefor
Example:
Example:
If may be necessary to solve equations involving the composition of two functions, for example
Example:Solve
]]>Definition: A stationary point is a point on a curve where
Definition: A stationary point, withis a minima ifOn a graph a minimum is lower than the points on either side as on the graph below.
Definition: A stationary point, withis a Maxima if A maximum is above the points on either side of it.
We often have to find the stationary points and classify them as maxima or minima. To do this,
We findand solveto find the x – coordinates of the turning points.
We put these values of x into the equation of the graph to find the y – coordinates of the tuning points.
We differentiateto findWe put the values of x from 1. into the expression forto find a value: if this value is positive then the stationary point is a minimum. If this value is negative then this stationary point is a maximum. If this value is zero then we can find y values a little either side of the stationary point and do a 3 point sketch to determine if a maximum, minimum.
Example:
Find the turning points and determine their nature.
This is shown below.
Example: Find the area under the curvebetweenand
To evaluate this integral we use the identityrearranging it to give
]]>Givento findwe differentiateTo findwe differentiateWe can differentiate any number of times: if we differentiate y n times, we have
Examples of differential equations include:
If a functionis a solution to the differential equation then we can substitutefor into the equation and obtain an identity.
Example: Show thatsatisfies the equation
Hence the given function satisfies the equation.
Example: Show thatsatisfies the equation
Hence the given function satisfies the equation.
]]>
Function |
||
We differentiateandand substitute them into The Chain Rule:
Example: Differentiate
Now just multiply the differentiated terms:
Our final answer must be in terms ofHence we substituteThe final answer is
Example:Differentiate e^(2x-1)
Now just multiply the differentiated terms:
Our final answer must be in terms ofHence we substituteThe final answer is
Example: Differentiate 1 over {x^2+2}
Now just multiply the differentiated terms:
Our final answer must be in terms ofHence we substituteThe final answer is
]]>