Help~ Don't remember what the point of tengency is.

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In summary, the problem involves finding a tangent line to the graph of f(x)= x3- 18x2+ 105 x- 146, passing through the point (2,0) and a point to the right of the low point at x=7. The slope of the tangent line can be found by setting up two equations involving the slope and the point of tangency, and solving for the variables.
  • #1
Here is the problem.

f(x)=x^3-18x^2=105x-146 and 2 is a real zero.

now, through the point (2,0), draw a line that is tangent to the graph at a point to the right of the low point at x=7. find the slope of the tangent line using x=2 and the point of tangency.

and here is the graph.
 

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  • #2
baby garfield said:
f(x)=x^3-18x^2=105x-146 and 2 is a real zero.
I assume that is f(x)= x3- 18x2+ 105 x- 146.
You want to find a line that passes through (2, 0) and is tangent to the graph at some (x,y) where x> 7.

Okay: any line through (2, 0) can be written as y= m(x-2) where m is the slope.

If the line is tangent to the graph at, say x1 , then it must satisfy the equation at that point: we must have m(x1- 2)= x13- 18x12+ 105 x1- 146.
It also must have the same slope there so m= f '(x1)= 3x12- 36x1+ 105.

Solve those two equations for m and x1.
 
  • #3


The point of tangency is the point where a line touches a curve at only one point, without crossing over or intersecting the curve. In this problem, the point (2,0) is a real zero of the function f(x)=x^3-18x^2-105x+146. This means that when x=2, the value of the function is equal to 0.

To find the slope of the tangent line at the point (2,0), we can use the formula for slope, which is rise over run. In other words, the slope is equal to the change in y over the change in x.

To find the change in y, we can use the given point (2,0) and the point of tangency (x,y). The change in y is equal to y-0, or simply y.

To find the change in x, we can use the given point (2,0) and the point of tangency (x,y). The change in x is equal to x-2.

Now, we can set up the equation for slope using these values:

slope = (y-0)/(x-2)

To solve for the slope, we need to find the coordinates of the point of tangency. From the graph, we can see that the low point of the curve is at x=7. This means that the point of tangency must be to the right of x=7.

To find the coordinates of the point of tangency, we can plug in x=7 into the original function:

f(7)=7^3-18(7)^2-105(7)+146=0

This means that the point of tangency is (7,0).

Now, we can plug in the values into the equation for slope:

slope = (0-0)/(7-2) = 0

Therefore, the slope of the tangent line at the point (2,0) is 0. This means that the tangent line is horizontal at this point.

To draw the tangent line, we can plot the points (2,0) and (7,0) and draw a straight line connecting them. This line will be tangent to the curve at the point (7,0).

In summary, the point of tangency is the point where a line touches a curve at only one point without crossing over or intersect
 

What is the point of tangency?

The point of tangency is the point where a tangent line touches a curve or surface at one single point. It is also the point where the slope of the tangent line is equal to the slope of the curve or surface at that point.

How is the point of tangency calculated?

The point of tangency can be calculated using the derivative of the function at that point. The derivative represents the slope of the curve at any given point, and setting it equal to the slope of the tangent line will give the coordinates of the point of tangency.

What is the significance of the point of tangency?

The point of tangency is important because it helps us determine the behavior of a curve or surface at a specific point. It also allows us to find the equation of the tangent line, which can be useful in various applications such as optimization problems.

Can there be more than one point of tangency on a curve?

Yes, there can be multiple points of tangency on a curve. This occurs when the curve changes direction or curvature at different points, resulting in multiple tangent lines that touch the curve at those points.

How is the point of tangency used in real-life applications?

The concept of the point of tangency is used in various fields such as physics, engineering, and economics. For example, in physics, it is used to determine the velocity and acceleration of an object moving along a curved path. In economics, it can help determine the maximum or minimum point of a function, which is useful in cost-benefit analysis and optimization problems.

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