Let f be a differentiable function such that f(3) = 2.345 and f' (x) = ln (x^2+ 1). What is the value of f (5)? A) 3.301
B) 5.631
С) 6.950
D) 7.976

Here we want to solve the differential equation, by finding the Taylor expansion of the function and evaluation we will find that f(5) = 6.950, so the correct option is C.

Here we must start by doing a Taylor expansion of the function, remember that the Taylor expansion is given by:

f(x) = f(a) + f'(a)*(x - a) + ...

The series has more terms, but we can work with two to get an estimation, if we use a = 3 we get:

f(x) = f(3) + ln(3^2 + 1)*(x - 3) = 2.345 + ln(10)*(x - 3)

Now we evaluate this in x = 5 to get:

f(5) = 2.345 + ln(10)*(5 - 3) = 6.950

So the correct option is C.

If you want to learn more about Taylor series, you can read:

link

Use a linear approximation to centered at :

Then

which makes C the most likely choice.

let soccer = x and golf = y

part a

1st equation: x + y = 125

2nd equation: x = y +45

part b

replace x in the 1st equation with the 2nd one:

y + 45 + y = 125

simplify:

2y + 45 = 125

subtract 45 from each side:

2y = 80

divide both sides by 2:

y = 80/2

y = 40

she plays golf for 40 minutes.

part c

no because she plays soccer for 45 minutes longer than golf 40 +45 =   85 minutes, not 80 minutes.