Car Value A family buys a car for 20,000The value of the car decreases about 20% each year. After 6 yr, the family decides to se
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Step-by-step explanation:
To answer your question fully can literally take up an entire book.
The importance becomes obvious if you consider the derivative (and the inverse of it the integral) as the foundation of calculus. I guess that askes the question what is calculus? For this answer a very very simplistic answer would be that it is the branch of mathematics that studies "non-constant" change (things that curve)
...... Let's see if this makes sense? You are driving your car at a constant 50 mph . after one hour you are 50 miles from home and after two you are 100.
y=50x. you can calculate this value in a several of ways just take two time stamps and two distance stamps ... subtract the two times and two
ok now here is the REAL PROBLEM issue we want to know our CHANGE IN
VELOCITY (that is ACCELERATION) ... if you take ANY two points you will
see that the SPEED is ALWAYS the same so the numerator is always 0 (zero) and thus the denominator (the difference in the time stamps) does not matter
now lets say that we are accelerating ..we are going y = 10 mph
at 3 seconds we are going 90 mph (10* 3^2) .. but we really don't care about that we want the "acceleration" at EXACTLY that point...
to do that we "have to estimate" delta Y over delta X
lets take 3 and 0 .... (9-0)/3 estimate of acceleration is 3...
but WAIT we can do better (get closer) take 3 and 1 ... (9 - 1)/2
acceleration is 4... once again lets get closer to the exact 3 ...
3 and 2 ... (9 -4)/2 the acceleration is 5...
hold on ! lets get even closer use 3 and 3 and we get it exactly !
(9-9)/0 that is the answer hurray! HOLD ON ! you CAN NOT DIVIDE BY ZERO ... no can do... we have to wait for newton and Leibniz to
figure out lim as x-> 0 f(x)-f(x+ delta x) / x to get around the DIVIDE BY ZERO problem...
that is the derivative
Step-by-step explanation:
The binomial theorem states that for some a,b∈R and some k ∈Z+ ,
(a+b)k=∑n=0k(kn)ak−nbn.
The binomial series allows us to use the binomial theorem for instances when k is not a positive integer. The binomial series applies to a given function f(x)=(1+x)k for any k∈R with the condition that |x|<1 . It is stated as follows:
(1+x)k=∑n=0∞(kn)xn .
Note that the binomial theorem produces a finite sum and the binomial series produces an infinite sum.