Question #bdf2d

2 Answers
Jan 6, 2018

#e^1.6=4.9530324244~~4.9530#

Explanation:

You can use a normal calculator, or draw the graph of #y=e^x# on a piece of paper.

graph{e^x [-0.427, 4.573, 2.74, 5.24]}

As you can see here, at #x=1.6#, #y# is around #4.95# or #4.96#. But, you will have to use a calculator to be more accurate.

Jan 6, 2018

Alternate way...

#e^(8/5) approx 230771 / 46875 approx 4.9231#

This is just a pure approximate, but actually isnt too close to the actually answer, see other answer for calculator value...

Explanation:

We can use the well-known fact that...

#e^x = 1/(0!) + x/(1!) + x^2/(2!) + x^3/(3!) + ... #

#e^x = sum_(r=0) ^(oo) x^r / (r!) #

We can use the first few terms of #e^x# to approximate #e^1.6#

We know #1.6 = 16/10 = 8/5 #

#e^(8/5) approx 1 + (8/5) + (8/5)^2 / 2 + (8/5)^3 / 6 + (8/5)^4 / 24 + (8/5)^5 / 120#

#=> e^(8/5) approx 1 + 8/5 + 32/25 + 256/375 + 512/1875 + 4096/46875#

Simplifying to yield, by finding the common denominator...

# e^(8/5) approx 46875/46875 + 75000/46875 + 60000/46875 + 32000/46875 + 12800 /46875 + 4096/ 46875 #

#=> color(orange)(e^(8/5) approx 230771 / 46875 approx 4.9231#

This is an approximate value...