Find line tangent to curve which is parallel to other line

In summary, to find a line tangent to a curve, you need to identify the point on the curve where you want the tangent line to be tangent to, use the derivative to find the slope of the tangent line, and then use the point-slope formula to find the equation of the tangent line. The derivative of a curve represents the slope of the curve at a specific point, which is calculated using various rules. Two lines are parallel if they have the same slope, and there can only be one line tangent to a curve at a given point. Finding a line tangent to a curve is significant in calculus and other fields of mathematics as it allows us to make predictions and understand the behavior of a curve.
  • #1
bubblescript
14
2

Homework Statement


Find the line tangent to the curve f(x)=0.5x2+3x-1 which is parallel to the line g(x)=x/2+0.5

Homework Equations


f'(x)=x+3

The Attempt at a Solution


I know it involves taking the derivative of f(x) and using it somehow, but I don't know where to go from there.
 
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  • #2
bubblescript said:

Homework Statement


Find the line tangent to the curve f(x)=0.5x2+3x-1 which is parallel to the line g(x)=x/2+0.5

Homework Equations


f'(x)=x+3

The Attempt at a Solution


I know it involves taking the derivative of f(x) and using it somehow, but I don't know where to go from there.

You want to find the tangent line to the curve ##f(x)##, parallel to the line ##g(x)##. If the tangent line has to be parallel with ##g(x)##, what do you know about the slope of this line?
 
  • #3
Both lines will need to have the same slope, 1/2.
 
  • #4
bubblescript said:
Both lines will need to have the same slope, 1/2.

Correct. Now, the derivative gives the slope of the tangent line in each point. Where is the slope ##1/2##?
Can you find the equation of the tangent line then?
 
  • #5
The slope is 1/2 when f'(x)=1/2:
1/2=x+3
x=-5/2
 
  • #6
bubblescript said:
The slope is 1/2 when f'(x)=1/2:
1/2=x+3
x=5/2

Minor mistake in your arithmetic.

You have a slope and a point (if you fix the mistake). There is a unique line that fits those 2 conditions. Can you find the equation?
 
  • #7
Yes caught the mistake.

So using -5/2 on f(x) gets f(-5/2)=-43/8, which means a point on the line is (-5/2, -43/8).
If the line is of the form y=1/2*x+b, we substitute the point for x and y:
-43/8=1/2*-5/2+b
b=-33/8
The line is: y=1/2*x-33/8
 
  • #8
bubblescript said:
Yes caught the mistake.

So using -5/2 on f(x) gets f(-5/2)=-43/8, which means a point on the line is (-5/2, -43/8).
If the line is of the form y=1/2*x+b, we substitute the point for x and y:
-43/8=1/2*-5/2+b
b=-33/8
The line is: y=1/2*x-33/8

That's correct! Well done!
 
  • #9
Thread moved. @bubblescript, please post problems that involve derivatives in the Calculus & Beyond section.
 
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FAQ: Find line tangent to curve which is parallel to other line

How do I find a line tangent to a curve?

To find a line tangent to a curve, you will need to first identify the point on the curve where you want the tangent line to be tangent to. Then, you can use the derivative of the curve at that point to find the slope of the tangent line. Finally, you can use the point-slope formula to find the equation of the tangent line.

What is the derivative of a curve?

The derivative of a curve represents the slope of the curve at a specific point. It is calculated by finding the limit of the change in the y-values over the change in the x-values as the change in x approaches 0. This can be simplified using the power rule, product rule, quotient rule, and chain rule.

How do I know if a line is parallel to another line?

Two lines are parallel if they have the same slope. This means that they have the same rate of change and will never intersect. You can compare the slopes of the two lines to determine if they are parallel.

Can there be more than one line tangent to a curve at a given point?

No, there can only be one line tangent to a curve at a given point. This is because the tangent line represents the instantaneous rate of change of the curve at that point, and there can only be one instantaneous rate of change at a specific point.

What is the significance of finding a line tangent to a curve?

Finding a line tangent to a curve is important in calculus and other fields of mathematics. It allows us to approximate the behavior of a curve at a specific point and make predictions about the curve's future behavior. It also helps us to understand the rate of change of a curve and its relationship to other lines or curves.

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