Find the maximum area of a triangle formed in the first quadrant by the x -axis, y -axis and a tangent line to the graph of f=(x+10)^-2.
f'(x) = -2/(x+10)^3
So, the tangent line through (h,k) on the curve is
y - 1/(h+10)^2 = -2/(h+10)^3 (x-h)
Find the intercepts of that line (a,0) and (0,b)
and the area is A = ab/2
Now find where dA/dh = 0 for maximum area
To find the maximum area of the triangle formed in the first quadrant by the x-axis, y-axis, and a tangent line to the graph of f = (x+10)^-2, we need to find the point on the graph where the tangent line is drawn.
Step 1: Find the derivative of f(x):
f'(x) = -2(x+10)^-3
Step 2: Set the derivative equal to zero to find the critical points:
-2(x+10)^-3 = 0
Since a negative number raised to a negative power does not equal zero, this equation has no real solutions. Therefore, there are no critical points on the graph.
Step 3: Find the equation of the tangent line at any random point on the graph.
Let's take the point P on the graph as (-5, f(-5)):
f(-5) = (-5+10)^-2 = 1/25
Step 4: Find the slope of the tangent line at the point P using the derivative:
m = f'(-5) = -2(-5+10)^-3
m = -2/125
Step 5: Use the point-slope formula to find the equation of the tangent line y = mx + b:
1/25 = (-2/125)(-5) + b
1/25 = 2/25 + b
b = -1/25
Therefore, the equation of the tangent line is y = (-2/125)x - 1/25.
Step 6: Find the x-intercept of the tangent line:
Setting y = 0 in the equation of the tangent line:
(-2/125)x - 1/25 = 0
-2/125 - 1/25 = 0
-1/25 = (2/125)x
x = -2/5
Step 7: Find the y-intercept of the tangent line:
Setting x = 0 in the equation of the tangent line:
y = (-2/125)(0) - 1/25 = -1/25
Step 8: Determine the area of the triangle formed by the tangent line and the x-axis and y-axis.
The base of the triangle is the distance between the x-intercept and the origin: |-2/5 - 0| = 2/5.
The height of the triangle is the distance between the y-intercept and the origin: |-1/25 - 0| = 1/25.
Therefore, the area of the triangle is (1/2) * base * height:
A = (1/2) * (2/5) * (1/25)
A = 1/125
Therefore, the maximum area of the triangle is 1/125 square units.
To find the maximum area of a triangle formed in the first quadrant by the x-axis, y-axis, and a tangent line to the graph of f=(x+10)^-2, we need to follow these steps:
Step 1: Understand the problem
We want to find the maximum area of a triangle formed in the first quadrant. The triangle's three sides are the x-axis, y-axis, and a tangent line to the graph of f=(x+10)^-2.
Step 2: Visualize the triangle
First, let's visualize the triangle by plotting the graph of f(x)=(x+10)^-2. Since the triangle is formed in the first quadrant, we are only interested in the positive x and y values.
Step 3: Find the tangent line
To find the tangent line, we need to differentiate the function f(x)=(x+10)^-2 with respect to x. Let's differentiate it using the power rule:
f'(x) = -2(x+10)^-3 * 1
Simplifying further:
f'(x) = -2 / (x+10)^3
Now, let's find the slope of the tangent line at a specific point on the graph. To do this, we can substitute the x-coordinate of the point into f'(x). Let's find the slope at some point (x, y) on the graph.
Step 4: Determine the triangle's base and height
To find the base and height of the triangle, we need to find the x and y-intercepts of the tangent line.
- The x-intercept is the point where the slope is zero. Let's solve for x when f'(x) = 0:
0 = -2 / (x+10)^3
Since the denominator cannot be zero, there are no x-intercepts.
- The y-intercept is the point where the tangent line intersects the y-axis. We can find it by substituting x = 0 into the equation of the tangent line:
y = f(0) = (0 + 10)^-2 = 1/100
So, the y-intercept is (0, 1/100).
Step 5: Calculate the area of the triangle
The area of the triangle can be calculated as A = (base * height) / 2. In this case, the base is the x-coordinate where the tangent line intersects the x-axis, and the height is the y-coordinate where the tangent line intersects the y-axis.
From Step 4, we found that the y-intercept of the tangent line is (0, 1/100). This gives us the height of the triangle.
To find the base, we need to consider the tangent line. Since the slope of the tangent line is given by f'(x), we substitute the x-coordinate of the y-intercept (0, 1/100) into f'(x):
f'(0) = -2 / (0+10)^3 = -1/500
The reciprocal of the slope gives us the base of the triangle:
base = 1/(-1/500) = -500
Since we are interested in positive values, we take the absolute value: base = 500.
Finally, we can now calculate the area of the triangle:
Area = (base * height) / 2 = (500 * 1/100) / 2 = 5
Therefore, the maximum area of the triangle formed in the first quadrant is 5 square units.