Tangent Line Problem: Find Point P & Compute Slope m_L

In summary, the conversation discusses finding the coordinates of a point P and the slope of a line L that is tangent to the graph of a given function. The first step is to find the derivative of the function, and then use the point-slope form to determine the coordinates of P. The slope of L can be found by equating the slope of the line from (0, 0) to (a, f(a)) with the slope of the tangent at any point on the curve.
  • #1
CrossFit415
160
0
I'm on mobile so I can't use latex.

Let C: y=8x^5+5x+1 and suppose L is a line through the origin tangent to C at a point P=(a,f(a)) on C.

-Find the coordinates of P

-Compute the slope m sub L of L

Where should I begin? I'm guessing I would need the derivative of the equation f(x)? Then I would use the point slope form to figure out point P?
 
Physics news on Phys.org
  • #2
CrossFit415 said:
I'm on mobile so I can't use latex.

Let C: y=8x^5+5x+1 and suppose L is a line through the origin tangent to C at a point P=(a,f(a)) on C.

-Find the coordinates of P

-Compute the slope m sub L of L

Where should I begin?
Maybe a sketch of the function?
CrossFit415 said:
I'm guessing I would need the derivative of the equation f(x)?
One would think that might somehow enter into things.
CrossFit415 said:
Then I would use the point slope form to figure out point P?
 
  • #3
After I sketch the eqiation what would I do after? So does that mean L is a secant line through the equation?
 
  • #4
L is a line that is tangent to the graph of the function. It's not a secant line (a line that hits a curve at two points).

BTW, "a secant line through the equation" doesn't make much sense, unless you actually draw a line through an equation, as opposed to drawing a line through the graph of an equation.
 
  • #5
Mark44 said:
L is a line that is tangent to the graph of the function. It's not a secant line (a line that hits a curve at two points).

BTW, "a secant line through the equation" doesn't make much sense, unless you actually draw a line through an equation, as opposed to drawing a line through the graph of an equation.

Haha I'll be more precise. I misread I thought it was going through the graph of the equation thinking its a secant line.

So I got the derivative which is f'(x) = (40x^4) + 5.

I wouldn't be able to apply the slope formula since I'm figuring out the coordinates of P. What methods are there?
 
  • #6
CrossFit415 said:
suppose L is a line through the origin tangent to C at a point P=(a,f(a)) on C.

So you have a line from (0, 0) to (a, f(a)) on the graph of your curve. Write an expression that represents the slope of the line.

Write another expression that represents the slope of the tangent at any point on your curve.

Equate the two expressions.
 

Related to Tangent Line Problem: Find Point P & Compute Slope m_L

1. What is the Tangent Line Problem?

The Tangent Line Problem is a mathematical problem that involves finding the equation of a line that is tangent to a given curve at a specific point. The goal is to determine the coordinates of the point of tangency and the slope of the tangent line.

2. How do you find the point of tangency?

The point of tangency can be found by first finding the derivative of the given curve. Then, set the derivative equal to the slope of the tangent line and solve for the variable. This will give you the x-coordinate of the point of tangency. To find the y-coordinate, plug in the x-coordinate into the original equation of the curve.

3. What is the slope of the tangent line?

The slope of the tangent line is equal to the derivative of the given curve at the point of tangency. It represents how steep the curve is at that particular point.

4. What is the difference between a tangent line and a secant line?

A tangent line is a straight line that touches a curve at only one point, while a secant line is a line that intersects a curve at two points. The tangent line represents the instantaneous rate of change at a specific point on the curve, while the secant line represents the average rate of change between two points on the curve.

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

Yes, there can be multiple tangent lines to a curve at a given point. This occurs when the curve has a sharp turn or inflection point, which means that the slope of the curve is constantly changing at that point.

Similar threads

Can a Function Have Two Different Tangent Lines at the Same Point?
    • Calculus and Beyond Homework Help
Replies
2
Views
805
Find the two points on the curve that share a tangent line
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Find Values of Osculating Circle Tangent to a Parabola
    • Calculus and Beyond Homework Help
Replies
4
Views
278
How can I overcome the lack of support for my passion for mathematics?
    • Calculus and Beyond Homework Help
Replies
15
Views
1K
Can you clarify your understanding for parts (a) and (b)?
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Find the slope of the tangent line to the function's inverse
    • Calculus and Beyond Homework Help
Replies
13
Views
3K
Finding the Slope of Secant and Tangent Lines
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Write the tangent lines to the curve
    • Calculus and Beyond Homework Help
Replies
1
Views
906
Find the equation of a tangent line to y = x^2?
    • Calculus and Beyond Homework Help
Replies
11
Views
4K
Equations of Parallel Tangent Lines to f(x)=3x(5x^2+1)
    • Calculus and Beyond Homework Help
Replies
11
Views
1K

Hot Threads

Recent Insights

Back
Top