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Andrea (Parent) May 19 2017
Phip (Parent) May 11 2017
Eddie Thomson (Student) May 9 2017
Eddie Thomson (Student) April 29 2017
Finding a maximum and minimum involves differentiating the function in order to determine when the gradient is at zero as this will be when the function has its maximum and minimum turning points.
take the function: f(x)=x^{3}+6x^{2}+9x+2
we must first obtain the gradient function through differentiation
d/dx = 3x^{2}+12x+9 using standard differentiation technique (multiplying the power of the unknown by its coefficient to generate its new coefficient and then reduce its power by one to find the new term)
now that we have the gradient function it must be set to zero, since the gradient of the function is zero at a turning point
3x^{2}+12x+9=0
this is a quadratic and we must solve for x to find the x values at which the gradient is zero
by inspection:
(3x+3)(x+3)=0 and therefore when the gradient of the function is zero:
x=1 and x=3
to determine which of these is maximum and is which is minimum we can differentiate the gradient function again. This generates a function that tells us the rate of change of the gradient at any given value of x
d/dx=3x^{2}+12x+9
d^{2}/dx^{2}=6x+12 (using the same method as before)
if we substitue in our two x values:
when x=1 d^{2}/dx^{2}= 6+12 = 6
since this is positive, the gradient is changing in a positive direction and so when x= 1 it is a minimum.
when x=3 d^{2}/dx^{2}= 18+12 = 6
since this is negative the gradient is changing in a negative direction and so when x= 3 its a maximum.
to find the y values of these x values we simply subsitute them into the original function:
f(x)=x^{3}+6x^{2}+9x+2
f(1)=1+69+2=2
f(3)=27+5427+2=2
so our maximum is (3,2)
and our minimum is (1,2)
see moresimulatneous equations are solved by eliminating one of the unknowns such as x or y. there are a number of ways to do this such as substitution or subtraction/addition. For this example we will use subtraction addition as i always found that the easier method at GCSE since it often avoids fractions.
e.g. 2y=3x+12 and 3y=5x+14
we need to choose one of the two unknowns to eliminate and then find their lowest common denominator. I will choose x, where the LCD is 15 (3*5=15). now we multiply the entire equation through in order that the number in front of x (called the coefficient) is 15. in the first equation, we multiply through by 5.
(2*5)y=(3*5)x+(12*5)
10y=15x+60
in the second equation we multiply through by 3 like before
(3*3)y=(5*3)x+(14*3)
9y=15x+42
now both the coefficients of x are 15 so we can subtract one equation from the other
(109)y=(1515)x+(6042)
y=0x+18
y=18
we now have the y value, since x is now zero. to find the x value all we need to do is go back to one of the first equations and put the value y=18 into the equation to resolve for x.
(2*18)=3x+12
3612=3x
24=3x
x=8
and thats everything. By making sure we can subtract one equation from another to eliminate one of the unkowns we can subsequently obtain both values. in this case, y=18 and x=8
see moreTotal resistance in a circuit depends on whether the resistors are in series or parallel since this determines which equation is to be used.
In series, resistors are essentially lined up next to each other so the total is simple since it is just the sum of the resistors resistance in ohms.
e.g. if a circuit has one 10ohm, one 45ohm and one 235ohm resistor in it, all three of which are in series, the total resistance is (10+45+235)ohm = 290ohms
In parallel each resistor is on its own path for the electricity to follow and as a result the equation is not quite the same, we use the equation:
1/R_{total }= 1/R_{1} + 1/R_{2 }+ 1/R_{3 }..etc
e.g. a circuit has 3 resistors all in parallel. Their resistances are 25ohm, 5ohm and 100ohm respectivley, what is the total resistance of the circuit?
1/R_{t} = 1/25 + 1/5 + 1/100
1/R_{t }= 4/100 + 20/100 + 1/100
1/R_{t }= 25/100
100 = 25R_{t}
R_{t }= 4ohms
often the answer will be a fraction when calculating a parallel due to the fractions involved in the total calculation however i used a simpler calculation for the example to demonstrate the equation's use
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