Tangent Lines, Normal Lines, and Rectilinear Motion

1. Tangent Lines, Normal Lines, and Rectilinear Motion Conner Moon Evan Haight 2nd period Mrs. Autrey

2. Tangent Lines • The definition of a tangent line is a line which intersects a curve at a point where the slope of both the curve and the line are equal. In other words, it is a line that touches a curve at one point without crossing over.

3. Three Things Needed to Find the equation of a tangent line • Derivative • Derivative at the point • Y-value at the point

4. Derivative • The first step of finding the equation of a tangent line is to take the derivative of the original equation.

5. Derivative at the point • The second step is to find the derivative at the point. For this example we will give the point x=2.

6. Cont… After finding the derivative at the point, your answer will be the slope of the tangent line.

7. Y-value at the point • Now go back to the original equation and plug in your x-value given earlier, x=2, to find the y-value.

8. Now, Slope Intercept Formula • After finding the slope of the tangent line, and your y-value, we plug in the those points to find the equation of the tangent line, shown below.

9. Example 1 Here we will find the equation of the tangent line at the point x=-2.

10. Walk Through

11. Solution

12. Try Me!!!! • Find the equation of the line tangent at the point x=-1

13. How did you do?

14. Normal Line • The definition of as normal line is a line that is perpendicular to the tangent line at the point of tangency. In other words it is a line that intersects the tangent line with a slope that is the negative reciprocal of the slope of said tangent line.

15. Things Needed to Find Equation of Normal Line • Original Equation of the Tangent line • Slope of Tangent line • Negative Reciprocal

16. Negative Reciprocal • Ok this sounds a lot harder than it actually is. So if the slope of the tangent line is y’=4, then the slope of the normal line, the negative reciprocal of the slope of the tangent line, is y’=-1/4.

17. Example 1 • Here we will find the equation of the normal line at the point x=-2

18. Walk Through This is the part where we take the negative reciprocal of the tangent line, sooo, the slope of this normal line is 1/8.

19. Solution

20. Example 2 • Here we will find the equation of the normal line at the point y=-1

21. Try Me!!!!

22. How did you do?

23. Rectilinear Motion • In this case when we talk about rectilinear motion lets think of it as the motion of a particle, and the motion of this particle is illustrated by a given expression. Using this expression and derivatives we will be able to calculate the position, velocity, and acceleration of said particle.

24. Stages of Rectilinear Motion • X(t)=position • V(t) / X’(t)=velocity • A(t) / X’’(t)=acceleration

25. Explanation of a Sign Line • Staying with the particle idea, a sign line will visually show us the positive or negative value of a group of numbers, using this information we can determine if the particle is moving left or right, has a positive or negative velocity, or if the acceleration of the particle is positive or negative.

26. Example 1 • Find when the particle, expressed by the equation , is moving to the right.

27. Walkthrough

28. Solution • As the sign line shows the particle is moving right from .

29. Try Me!!!! • Find when the particle, expressed by the equation , is moving to the left.

30. How did you do??

31. Now you know how to do tangent and normal lines and rectilinear motion. The end.