Finding Perpendicular Tangent Point on Parabola

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SUMMARY

The discussion focuses on finding the point on the parabola defined by the equation y = 4x² + 2x - 5 where the tangent line is perpendicular to the line represented by 3x + 2y = 7. The slope of the given line is calculated to be -3/2, leading to a perpendicular slope of 2/3. By applying the derivative of the parabola, dy/dx = 8x + 2, the x-value where the slope equals 2/3 is determined to be -1/6. Substituting this x-value back into the parabola equation yields the corresponding y-coordinate.

PREREQUISITES
  • Understanding of derivatives and their applications in calculus.
  • Knowledge of the concept of perpendicular slopes in geometry.
  • Ability to manipulate algebraic equations and solve for variables.
  • Familiarity with the standard form of a quadratic equation.
NEXT STEPS
  • Study the concept of derivatives in calculus, focusing on their geometric interpretation.
  • Learn how to find the equation of a tangent line to a curve at a given point.
  • Explore the relationship between slopes of perpendicular lines in more depth.
  • Practice solving quadratic equations and their applications in real-world problems.
USEFUL FOR

Students studying calculus, particularly those focusing on derivatives and their applications, as well as anyone interested in understanding the geometric properties of parabolas and lines.

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Homework Statement


Find the point on the parabola y= 4x^2 + 2x - 5 where the tangent line is perpendicular to the line 3x + 2y = 7.


Homework Equations





The Attempt at a Solution


I don't know what to do since I was away the last 3 classes since I was away. Help me please.
 
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You want to find the slope of the line you're given and use the definition of a derivative as well as perpendicularity to solve for the x value you want.
 
Two perpendicular lines have slopes that are negative reciprocals of each other, eg: a line with a slope 2 is perpendicular to a line with a slope -1/2.

Find the slope of the line, find the negative reciprocal of that slope. The derivative of a function is the slope of that graph at any point on the graph, so find the derivative of the parabola and see at what value of x it will equal the negative reciprocal of the slope you found earlier.
 
TMM said:
You want to find the slope of the line you're given and use the definition of a derivative as well as perpendicularity to solve for the x value you want.

So I take the slope of this? 3x + 2y = 7

so...

2y = -3x + 7
y= -3/2x + 7/2

slope = -3/2 so if it is perpendicular the slope is 2/3 is that my right slope?

I now I have to do more but it that right so far?
 
Yes.
 
Correct.

dy/dx = 8x+2
You want the value of x when dy/dx is (2/3), as you said from above.
Solving for x gets (-1/6).
Plug this value into your original equation y=4x^2 etc.
 

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