We must find the equation of the tangent line for the function g(x) at the point x = 5, given:

The general equation of the tangent line is:

where:
• m is the slope,
,
• b is the y-intercept.
1) Slope of the line
To find the equation of the line, first, we must find the slope of the line at x = 5, which is given by the derivative of the function g(x) at x = 5.
The Fundamental Theorem of Calculus says that:

Using the graph of the function f, we get:

So the slope of the line tangent to g(x) at the point x = -1 is:

So the equation of the line has the form:

We must find the value of b.
2) y-intercept of the line
If the line is tangent to g(x) at the point with x = 5, the line must have the same value as g(x) at x = 5:

We must compute the value of g(5), which is given by:

So the value of g(5) is the area under the curve f(t). Summing the different contributions, we get:

Replacing the result in the equation above, we get:

Using the value m = -1 and b = 3, the equation of the tangent line is:

Answer
