Use the first five terms of the exponential series and a calculator to approximate each value to the nearest hundredth.
e^3.5
I honestly don't understand the concept or how to solve the problem. :(
e^x = 1+x+x^2/2!+x^3/3!+x^4/4! ......
No worries! I'll explain how you can approximate the value of e^3.5 using the first five terms of the exponential series and a calculator.
To solve this problem, we'll use the Taylor series expansion of the exponential function. The Taylor series expansion for e^x is given by:
e^x = 1 + x + (x^2 / 2!) + (x^3 / 3!) + (x^4 / 4!) + ...
For our specific problem, we want to approximate e^3.5. We'll use the first five terms of the expansion to get an approximation.
Let's plug in the value of x as 3.5 into the Taylor series expansion. We'll evaluate the first five terms and sum them up.
1st term: 1
2nd term: x = 3.5
3rd term: (x^2 / 2!) = (3.5^2 / 2)
4th term: (x^3 / 3!) = (3.5^3 / 3!)
5th term: (x^4 / 4!) = (3.5^4 / 4!)
Summing up these five terms will give us an approximation of e^3.5.
Now let's use a calculator to calculate the approximate value using the first five terms of the expansion:
1. Square 3.5: 3.5^2 = 12.25
2. Divide 12.25 by 2: 12.25 / 2 = 6.125
3. Cube 3.5: 3.5^3 = 42.875
4. Divide 42.875 by 6 (3 factorial): 42.875 / 6 = 7.1458
5. Raise 3.5 to the power of 4: 3.5^4 = 150.0625
6. Divide 150.0625 by 24 (4 factorial): 150.0625 / 24 = 6.2526
Finally, sum up all the terms:
1 + 3.5 + 6.125 + 7.1458 + 6.2526 = 24.0734 (approximation)
So, using the first five terms of the exponential series, e^3.5 is approximately 24.0734 when rounded to the nearest hundredth.
Remember, this is just an approximation. The actual value of e^3.5 is approximately 33.1155.