For a particular function g, we know g'(8) = 4 and g(8) = -5. Write an equation for the tangent to g at x=8.

Answer 1

The slope of a line tangent to a curve/function at a point is equal to the first derivative of the function at that point. Therefore, the slope of the tangent line we seek is 4 because g'(8)=4. This allows us to write the equation of the tangent line (using y-intercept form) as y=4x+b where b is the y intercept. We can calculate the y intercept by using the slope (4) and the point-slope formula because we know two points on the tangent line: (8,-5) [because g(8)=-5 and the line is tangent at that point] and (0,b) which is the y-intercept. Point-slope formula is (y₁-y₂)/x₁-x2₎=m. Plugging in the two points we know that lie on the line gives us

(-5-b)/(8-0)=4 so b=-37, and the tangent line's complete formula is y=4x-37.

Answer 2

Slope of tangent line = g'(8) = 4

Since g(8) = -5, the point (8, - 5) is on the graph of y = g(x)

(8, - 5) is the point where the tangent line intersects the graph of y = g(x).

Equation of tangent line: y - (-5) = 4(x - 8)

So, y = 4x - 37