A person places $26300 in an investment account earning an annual rate of 3.8%, compounded continuously. Using the formula V=Pe∧
2 answers:
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Answer:
h=0.4--> 15
h=0.2 --> 14.06
h=0.1 --> 13.71
Step-by-step explanation:
This is a numerical solution using Euler's method. Euler's method enables us to numerically approch a solution with a suitable step size. As the step size gets smaller, the approximation will be more accurate. Euler's method is as the following.
Here, h is the step size. The reason why first derivation used here is to make an appriximation with using the rate of increase or decrease. As the step size is smaller, the icreases or decreases are followed more accurately. Now, let's solve the question:
for h= 0.4:
y(0.4)=y(0)+0.4*y'
The trick here is y' is equal to y, thus we can write that y'=y(0.4) and our starting point y(0) is given as 9 in the question and the equation becomes:
y(0.4)=9+0.4*y(0.4) and this is easy to solve.By replacing y(0.4) functions to be at the same side of the equation, we get:
0.6*y(0.4)=9 and by solving this equation, y(0.4) is found to be 15.
for h=0.2:
This will be similar to the previous question, but since the step size is 0.2, we will first calculate y(0.2) and then y(0.4).
y(0.2)=y(0)+0.2*y(0.2) and y(0) is 9.
0.8*y(0.2)=9 and y(0.2)=11.25. Now, we will replace this value into the next iteration o the formula istead of y(0). The equation is like:
y(0.4)=y(0.2)+0.2*y(0.4) and y(0.4) is found to be 14.06.
for h=0.1:
This is also similar to the above solutions but will be longer and have 4 iterations.
first iteration: y(0.1)=9+0.1*y(0.1) --> y(0.1)=10
second iteration: y(0.2)=y(0.1)+0.1*y(0.2) --> y(0.2)=11.11
third iteration: y(0.3)=y(0.2)+0.1*y(0.3) --> y(0.3)=12.34
fourth iteration: y(0.4)=y(0.3)+0.1*y(0.4) --> y(0.4)=13.71
As the step size gets smaller, the answer also gets smaller and more accurate. With even smaller step sizes, there will be a better approximation. However, in case you have more complex equations or smaller step sizes, it is recommended to use a computer software to make an approximation.
Answer:
Step-by-step explanation:
A function satisfying the equation is said to be an even function. This denomination comes from the fact that the same relation is satisfied for functions of the form
with
even. Observe that if
is twice differentiable we can derivate using the chaing rule as follows:
implies
Applying the chain rule again we have:
implies
So we have that function is also an even function.