If a figure shows a function g(x) and its tangent line at the point B=(2,6.8). If the point A on the tangent line is (1.94,6.87), fill in the blanks below to complete the statements about the function g at the point B.
g(___) = ______
g(___) = ______
To complete the statements about the function g at the point B, we need to determine the values for the functions g(x) and g'(x) at x = 2.
The point B=(2,6.8) lies on the tangent line of the function g(x) at that point, which means the slope of the tangent line represents the derivative of g(x) at x = 2. Since we are given point A=(1.94,6.87) on the tangent line, we can use the slope formula to find the derivative:
slope = (Change in y) / (Change in x)
= (6.87 - 6.8) / (1.94 - 2)
= 0.07 / (-0.06)
= -1.1667
So, the derivative of g(x) at x = 2, denoted as g'(2), is -1.1667.
Now, to find g(x) at x = 2, we can use the point (1.94,6.87) and the slope of the tangent line to find the equation of the tangent line. The equation of the tangent line can be expressed in the point-slope form:
y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line and m is the slope of the line.
Using the values from point A, we can write the equation of the tangent line as:
y - 6.87 = -1.1667(x - 1.94)
Simplifying this equation gives us:
y - 6.87 = -1.1667x + 2.2683
Next, we substitute x = 2 into this equation, since we want to find g(2):
g(2) - 6.87 = -1.1667(2) + 2.2683
Solving this equation will give us the value of g(2):
g(2) - 6.87 = -2.3334 + 2.2683
g(2) - 6.87 = -0.0651
g(2) = 6.8049
So, the statements about the function g at the point B=(2,6.8) are:
g(2) = 6.8049
g'(2) = -1.1667