Find the value of the function y=sqrt(x+1)+sin(x)-0.5, correct to 3 decimal places, when x=0.05 without the use of a calculator.

y = sqrt (x+1) + sin x I assume radians not degrees - .5

= sqrt(1.05) + sin (.05) - .5

first the (1.05)^(1/2)
well if s = t^.5
ds/dt = .5 t^-.5
ds = .5 t^-.5 dt
s (t+dt) = s(t) + .5 dt/sqrt t
here t = 1 and dt = .05
so
sqrt(1.05) = 1 + .5(.05)/1
sqrt(1.05) = 1 + .025 = 1.025

now for small x, sin x = x
so sin .05 = .05

so
1.025 + .05 - .5
= .575

Thank you very much. This problem was giving me too much stress.

To find the value of the function y = √(x+1) + sin(x) - 0.5 when x = 0.05 without a calculator, we can follow these steps:

1. Substitute the given value x = 0.05 into the function:
y = √(0.05+1) + sin(0.05) - 0.5

2. Simplify the expression inside the square root:
y = √(1.05) + sin(0.05) - 0.5

3. Calculate the value of sin(0.05):
sin(0.05) ≈ 0.05 (using the small angle approximation)

4. Substitute the value of sin(0.05) into the expression:
y = √(1.05) + 0.05 - 0.5

5. Simplify the expression inside the square root:
y = √0.05 + 0.05 - 0.5

6. Evaluate the square root:
√0.05 ≈ 0.223 (using estimation or by looking up the value in a table)

7. Substitute the value of √0.05 into the expression:
y = 0.223 + 0.05 - 0.5

8. Combine like terms:
y ≈ 0.273 - 0.5

9. Evaluate the subtraction:
y ≈ -0.227

So, the value of the function y = √(x+1) + sin(x) - 0.5 when x ≈ 0.05, correct to 3 decimal places, is approximately -0.227.