the tangent line to the graph of y=g(x) at the point (4, 5) passes through the point (7, 0) find g(4) and g'(4)
Well, let me start with a joke to lighten the mood! Why don't scientists trust atoms? Because they make up everything!
Now, onto the question. If the tangent line to the graph of y = g(x) passes through the points (4, 5) and (7, 0), we can find the slope of the line by using the formula:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) is (4, 5) and (x2, y2) is (7, 0).
So, substituting the values into the formula:
m = (0 - 5) / (7 - 4),
m = -5 / 3.
Now, g'(4) is equal to the slope of the tangent line at x = 4. Since we already have the slope (-5/3), g'(4) is -5/3.
To find g(4), we can use the fact that g(4) is equal to the y-coordinate of the point on the graph of y = g(x) when x = 4. Since the tangent line passes through (4, 5), we can conclude that g(4) = 5.
So, g(4) = 5 and g'(4) = -5/3. Keep smiling and ask me anything else you'd like!
To find g(4) and g'(4), we will use the given information that the tangent line passes through the points (4, 5) and (7, 0).
Step 1: Finding the slope of the tangent line
The slope of the tangent line can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values of the two points, we have:
m = (0 - 5) / (7 - 4)
m = -5 / 3
So, the slope of the tangent line is -5/3.
Step 2: Finding g'(4)
The value of g'(4) is the derivative of g(x) evaluated at x = 4. Since the slope of the tangent line at (4, 5) is equal to g'(4), we have g'(4) = -5/3.
Step 3: Finding g(4)
To find g(4), we need to use the point-slope form of a line and the equation of the tangent line:
y - y1 = m(x - x1)
Substituting the values of the point (4, 5) and the slope, we have:
y - 5 = (-5/3)(x - 4)
Expanding the equation, we get:
3y - 15 = -5x + 20
Rearranging the equation to solve for y, we have:
3y = -5x + 35
Dividing both sides by 3, we get:
y = (-5/3)x + 35/3
Therefore, g(4) is the y-coordinate of the point (4, 5) on the graph of g(x). Substituting x = 4, we can find g(4):
g(4) = (-5/3)(4) + 35/3
Simplifying this expression, we get:
g(4) = -20/3 + 35/3
g(4) = 15/3
g(4) = 5
To find g(4), we need to evaluate the function g(x) at x = 4. However, we are not given the function g(x) explicitly, so we have to use the information about the tangent line and the point (4, 5).
We know that the tangent line to the graph of y = g(x) at the point (4, 5) passes through the point (7, 0).
First, let's find the slope of the tangent line using the two given points. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Plugging in the coordinates of the two points, we have:
slope = (0 - 5) / (7 - 4) = -5/3
Now, since the tangent line has the same slope as the graph of g(x) at x = 4, we can find g'(4) by equating the slope to the derivative of g(x) at x = 4:
g'(4) = -5/3
So, the derivative of g(x) at x = 4 is -5/3.
To find g(4), we can use the point-slope form of a linear equation. The equation of a line with slope m passing through the point (x₁, y₁) is given by:
y - y₁ = m(x - x₁)
Plugging in the coordinates of the point (4, 5) and the slope -5/3, we have:
y - 5 = (-5/3)(x - 4)
Now, we can solve this equation to find y, which represents g(4):
y = (-5/3)(x - 4) + 5
Simplifying this equation, we have:
y = (-5/3)x + 20/3
Therefore, g(4) is equal to the value of the equation when x = 4:
g(4) = (-5/3)(4) + 20/3 = -20/3 + 20/3 = 0
So, g(4) is equal to 0. Therefore, g(4) = 0 and g'(4) = -5/3.
well g of 4 is 5 given
g' (4) = slope of tangent at (4,5)= (0-5)/(7-4) = -5/3