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(___) = ______
well, from the two points on the line you can get slope. Do that.
then you know g'(2)=slope that goes in one of the blanks.
You also know g(2)=6.8 put that in the second blank.
I tried and I got -1.17 for one of them... I plugged both answers and I just got the g(2) right, I'm not sure what to do now....
g(x) at the point B=(2,6.8) can be represented as:
g(2) = 6.8
g(x) at the point A=(1.94,6.87) can be represented as:
g(1.94) = 6.87
To complete the statements about the function g at the point B, we need to determine the x-coordinate and the corresponding y-coordinate of point B.
Given that point B lies on the tangent line to the function g(x) at (2, 6.8), we know that the tangent line at point B is the same as the function g(x) at that point. Therefore, the x-coordinate of point B is 2.
To find the y-coordinate of point B, we can use the fact that point A on the tangent line is (1.94, 6.87). Since point A lies on the tangent line, the slope of the tangent line at A is the same as the slope of the function g(x) at point B.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1) / (x2 - x1). Using this formula, we can calculate the slope of the tangent line at point A:
Slope = (6.87 - 6.8) / (1.94 - 2) = 0.07 / (-0.06) = -1.1667
Since the slope of the tangent line at point A is the same as the slope of the function g(x) at point B, we can use the point-slope form of a linear equation to find the equation of the tangent line:
y - y1 = m(x - x1)
where (x1, y1) is the point A and m is the slope of the tangent line:
y - 6.87 = -1.1667(x - 1.94)
Now, we can substitute the x-coordinate of point B (which is 2) into this equation:
y - 6.87 = -1.1667(2 - 1.94)
Simplifying the equation:
y - 6.87 = -1.1667(0.06)
y - 6.87 = -0.07
y ≈ 6.8
Therefore, the y-coordinate of point B is approximately 6.8.
Now, we can fill in the blanks:
g(2) = 6.8
So, g(x) = 6.8 and g(2) = 6.8.