Determine equations of the lines tangent to

In summary, the problem involves finding equations of tangent lines to the graph of y= x√(5-x²) at the points (1,2) and (-2,-2). To do so, we must first find the derivative of the function at the given points and then use the point-slope formula to determine the equations of the tangent lines. This is a fundamental concept in Calculus and understanding it is crucial for solving this problem.
  • #1
krete77
17
0
The problem is:
Tangent Lines: Determine equations of the lines tangent to the graph of Y = x√(5-x²) at the points (1,2) and (-2,-2). Graph the function and the tangent lines.

I have no IDEA where to go with this. I am taking calculus over the summer and we are in week 2 and I'm struggling..if anyone could do a step by step process here and explain I would be so grateful. Thanks in advance.
 
Physics news on Phys.org
  • #2
One of the first things you should have learned in Calculus is that the derivative of a function, at a given value of x, is the slope of the tangent line to the graph at that point on the graph. To find the slope of the tangent line at (2, 2) , find the derivative of [itex]y= x\sqrt{5- x^2}[/itex] at x= 2, then use the "point-slope" formula for the equation of the line having that slope through the line (2, 2).
 

1. What is the formula for finding the equation of a line tangent to a curve?

The general formula for finding the equation of a line tangent to a curve is y = mx + b, where m is the slope of the tangent line and b is the y-intercept. However, the specific formula can vary depending on the type of curve and the point of tangency.

2. How do you find the slope of the tangent line to a curve?

The slope of the tangent line can be found by taking the derivative of the function representing the curve at the point of tangency. This derivative will give the slope of the tangent line at that point.

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

Finding the equation of a line tangent to a curve allows us to determine the instantaneous rate of change of the curve at a specific point. This can be useful in many real-world applications, such as calculating velocity or acceleration in physics.

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

Yes, there can be multiple lines tangent to a curve at a given point. This typically occurs when the curve has a sharp turn or point of inflection at that point, and the slope of the curve changes abruptly.

5. How can I check if my equation of a line tangent to a curve is correct?

You can check the accuracy of your equation by plugging in the coordinates of the point of tangency into the equation and verifying that it satisfies the equation. Additionally, you can graph the equation and visually compare it to the curve at that point.

Similar threads

Quadratics Having a Common Tangent
    • Calculus and Beyond Homework Help
Replies
1
Views
126
Find eqn for "normal" line to the tangent- folium
    • Calculus and Beyond Homework Help
Replies
6
Views
983
Find the two points on the curve that share a tangent line
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Find the differential equation
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Finding the horizontal tan() lines of this equation
    • Calculus and Beyond Homework Help
Replies
3
Views
889
Calculating the equations for the tangent/normal lines
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Solve the attached problem that involves circle and tangent
    • Calculus and Beyond Homework Help
Replies
2
Views
598
Solving Tangent Lines for 2 Graphs: x1,x2
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Write the tangent lines to the curve
    • Calculus and Beyond Homework Help
Replies
1
Views
883
Find Values of Osculating Circle Tangent to a Parabola
    • Calculus and Beyond Homework Help
Replies
4
Views
110

Hot Threads

Recent Insights

Back
Top