Tangent lines parallel

In summary, the line x-2y+9=0 is tangent to the graph of y=f(x) at (3,6) and is also parallel to the line through (1,f(1)) and (5,f(5)). The slope of the first line is 1/2 and the slope of the line passing through (1,2) and (5, f(5)) is also 1/2. Using the definition of slope, we can solve for f(5) and the correct answer is (C) 4.
  • #1
syeh
15
0

Homework Statement


The line x-2y+9=0 is tangent to the graph of y=f(x) at (3,6) and is also parallel to the line through (1,f(1)) and (5,f(5)). If f is differentiable on the closed interval [1,5] and f(1)=2, find f(5)

A) 2
B) 3
C) 4
D) 5
E) None of these

The correct answer is (C) 4

The Attempt at a Solution



So I know the tangent line to (3,6) and f(5) have the same slope:

x-2y+9=0
2y=x+9
y=.5x+4.5

So, the slope is .5 at f(1), f(3), and f(5)

Now, I need to find the point at f(5). I do not know how to do this
 
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  • #2
1) The slope of the first line is 1/2

2) The slope of the line is passing through (1,2) and (5, f(5)). Then use the definition of slope:

[tex]1/2=\frac{f(5)-2}{5-1}[/tex]

and solve for f(5).
 

1. What is a tangent line?

A tangent line is a line that touches a curve at only one point. It is perpendicular to the radius of the curve at that point.

2. How do you determine if two tangent lines are parallel?

To determine if two tangent lines are parallel, you can use the slope of the lines. If the slopes are equal, then the lines are parallel. You can also visually inspect the lines to see if they have the same direction and do not intersect.

3. Can two tangent lines be parallel to a curve at different points?

Yes, it is possible for two tangent lines to be parallel to a curve at different points. This occurs when the curve has a point of inflection, where the tangent lines at that point have the same slope.

4. What is the significance of parallel tangent lines?

Parallel tangent lines indicate that the curve is either a straight line or a circle. In other words, it indicates that the curve has constant slope or constant radius at any point.

5. How can parallel tangent lines be used in real life applications?

Parallel tangent lines have various applications in physics, engineering, and geometry. For example, they can be used to determine the direction of motion of an object in a circular path or to find the rate of change of a function at a given point.

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