let f be the function whose graph goes through the point (3,6) whose derivatie is given by f'(x) =(1+e^x)/(x^2)

1. write an equation of a line tangent to the graph at x=3 and use it to approximate f(3.1)
--- I GOT : y= (1+e^3)/(9) (.1) +6

i don't know if it is right

i don't know how to do the second part thank you!

2. use integral [ from 3 to 3.1 to evaluate f(3.1)

To answer the first part of the question, you need to find the equation of the line tangent to the graph of the function f at x=3. The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

1. Calculate the slope of the tangent line:
To find the slope of the tangent line at x=3, you need to evaluate the derivative, f'(x), at x=3. Substitute x=3 into the given derivative equation:
f'(3) = (1+e^3)/(3^2)

2. Substitute the given point (3,6) into the equation y = mx + b:
6 = m(3) + b

3. Solve for the y-intercept, b, by rearranging the equation:
b = 6 - 3m

4. Substitute the value of m from step 1 into the equation from step 3 to find the y-intercept:
b = 6 - 3((1+e^3)/(3^2))

So, the equation of the line tangent to the graph of f at x=3 is y = (1+e^3)/(9)(x-3) + 6.

For the second part of the question, you need to use integration to evaluate f(3.1). Integration is the reverse process of differentiation and can be used to find the area under a curve.

1. Write the integral:
To evaluate f(3.1), you need to calculate the definite integral of f(x) from 3 to 3.1. The integral can be written as:
∫[3 to 3.1] (1+e^x)/(x^2) dx

2. Evaluate the integral:
To evaluate the integral, you can use various techniques such as Riemann sums, numerical methods, or software. If you have access to a computational tool or software that can perform integration, you can input the integral expression and evaluate it directly.

Alternatively, you can approximate the integral by using numerical methods such as Simpson's rule or the trapezoidal rule, which involve dividing the interval into smaller subintervals and approximating the area under the curve.

Once you evaluate the definite integral, the result will give you the approximate value of f(3.1).

Please note that the exact value of f(3.1) depends on the specific function f(x) and cannot be determined without further information.