LRAM is left hand approximation and the formula is:
delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
Say you are asked to calculate the left Riemann Sum for -4x -5 on the interval [-3, -1] divided into 2 subintervals.
delta x would equal: -1+3 /2 = 2/2 = 1
1[ f(-3) + f(-3 +1)]
1[ f( -3) + f(-2)]
then plug into your equation
RRAM is right hand approximation and the formula is:
delta x [ f(a + delta x) + .... + f(b)]
so using the same example:
1[ f( -2) + f(-1)] and then plug into your equation
MRAM is to calculate the middle and the formula is:
delta x [ f(mid) + f(mid) + .... ]
To find midpoints, you would add the two numbers together then divide by two
In this problem the numbers would be: -3 , -2, -1
-3 + -2/ 2 = -5/2 and -2 + -1 / 2 = -3/2
so 1[f(-5/2) + f(-3/2)] and the plug in
Trapezoidal is different because instead of multiplying by delta x, you multiply by delta x/2 and you also have on more term then your number of subintervals.
The formula is : delta x/2 [f(a) + 2f(a + delta x) + 2f(a+ 2 delta x) + ....f(b)]
For this problem: 1/2 [ f(-3) + 2 f(-2) + f( -1)] and then plug in.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
e integration:
whatever is raised to the e power will be your u and du will be the derivative of u. For example:
e^2x-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
related rates:
The steps for related rates are….
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
okay so ive been having trouble with trapizoidal and the particle problems....if anyone can explain please do cause im lost....
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The formula for trapezoidal is 1/2 delta x [f(a) + 2f(a+ delta x) +2f(a+2delta x) + .... f(b)]
ReplyDeleteNow when given the charts, you do not actually have to plug into the formula since the numbers are already given.
For 89, all you have to do is find delta x, which is done by seeing what they are counting by.
x = 0 0.5 1.0 1.5 2.0
f(x) = 3 3 5 8 13
.5 - 0 = 1/2
1-.5 = 1/2
and so on so delta x is 1/2
Now for trapezoidal, you need to divide that by 2
1/2/2/1 = 1/4
And now just plug in.
1/4 [3 + 2(3) + 2(5) + 2(8) + 13]
1/4 [3 +6+10+16+13}
1/4[48] = 12
The trapezoidal approximation is 12.
well for particle problems look at it as a list like
ReplyDeleteposition
velocity
acceleration
When moving from position down take the derivative and when moving from acceleration up integrate.