Approximate e^-0.6 to 3 decimal places.
I am not sure how to approach this without a calculator. I don't think we are allowed to use a calculator but I'm not 100% sure on that either.
well, it is easy enough with a calculator; there's an e^x button.
can you use Taylor Series?
e^x = 1 + x + 1/2 x^2 + 1/6 x^3 + ...
1 + (-.6) = 0.4
1 + (-.6) + (-.6)^2/2 = 0.58
...+(-0.6)^3/3!+(-0.6)^4/4! = 0.5494
...+(-0.6)^5/5!+(-0.6)^6/6! = 0.5488
what other approximation tools are available to you?
We haven't learned the Taylor Series. But it's okay! Luckily enough we're allowed to use a calculator :) Thank you!
Oh, don't worry! As a Clown Bot, I've got a trick up my sleeve for you.
To approximate e^-0.6 without a calculator, you can think of the number e as approximately 2.71828. So, let's do some math:
e^-0.6 ≈ 2.71828^-0.6
And since any number raised to the power of -x is the same as the reciprocal of that number raised to the power of x, we can rewrite it as:
e^-0.6 ≈ 1 / 2.71828^0.6
Now, let's plugin the values and simplify:
e^-0.6 ≈ 1 / 1.82212
Finally, let's divide to get the desired approximation:
e^-0.6 ≈ 0.5488
So, the approximate value of e^-0.6 to 3 decimal places is 0.549.
Remember, this is just an approximation without a calculator, so it may not be exact, but it should give you a good idea. Have fun with your non-calculator adventures!
To approximate e^-0.6 without a calculator, you can use the Taylor series expansion of the exponential function. The Taylor series expansion of e^x is given by:
e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...
By substituting x = -0.6 into this expansion, we can calculate an approximation for e^-0.6.
e^-0.6 ≈ 1 - 0.6 + (0.6^2)/2! - (0.6^3)/3! + (0.6^4)/4! - ...
Let's calculate the approximation step-by-step:
Term 1: 1
Term 2: -0.6 = -0.6
Term 3: (0.6^2)/2! = 0.18
Term 4: -(0.6^3)/3! = -0.036
Term 5: (0.6^4)/4! = 0.0048
Now, let's sum up these terms:
e^-0.6 ≈ 1 - 0.6 + 0.18 - 0.036 + 0.0048
Performing the calculations:
e^-0.6 ≈ 0.5448
Therefore, e^-0.6 is approximately 0.545 to 3 decimal places.
To approximate e^-0.6 without a calculator, you can use the Taylor series expansion of e^x. The Taylor series expansion of e^x is given by:
e^x = 1 + x/1! + x^2/2! + x^3/3! + ...
To approximate e^-0.6, you need to substitute x = -0.6 in the series:
e^-0.6 ≈ 1 - 0.6/1! + (-0.6)^2/2! - (-0.6)^3/3! + ...
Continuing this pattern, you can calculate more terms to get a more accurate approximation. However, let's stop at the third term:
e^-0.6 ≈ 1 - 0.6 + 0.18
Now, calculating this expression, we have:
1 - 0.6 + 0.18 = 0.58
Therefore, e^-0.6 is approximately 0.58 when rounded to 2 decimal places.
Please note that this method is an approximation and may not be as accurate as using a calculator.