Calculating the equations for the tangent/normal lines

In summary, the conversation discusses finding the slope and equation of a line given a point and a tangent line. The tangent slope of 0 corresponds to a vertical line with an equation of x=k, and the tangent slope of undefined corresponds to a horizontal line with an equation of y=k. For question 10, the equation would be x=-1 since f'(-1)=0. For question 12, the equation would be y=0 since the point given is (2,0) and the normal slope is 0.
  • #1
fulton33
3
0
Homework Statement
Use the table to write the equation for tangent lines at given values of x.
Relevant Equations
y-y0=m(x-x0)
IMG_3607.jpeg

9. When I do this problem I know my slope is -3 because f'(2)=-3. I then went and substituted and got
y+5=-3(x-2) which simplified to y=-3x+1

10. I get lost here because the tangent slope would be 0, which would give me the equation y=-2. The normal means perpendicular and the perpendicular slope to 0 is undefined. Not sure if that is right and what to do after.

11. I did the same steps in 9. The slope is 3 and I get the equation y-4=3(x+1) which simplifies to y=3x+7

12. I am lost here as well. The tangent slope would be DNE, which would mean the normal slope to be 0. When I plug 0 in for m and (2,0) for x and y I get y=0. I think that is wrong.
 
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  • #2
fulton33 said:
10. I get lost here because the tangent slope would be 0, which would give me the equation y=-2. The normal means perpendicular and the perpendicular slope to 0 is undefined. Not sure if that is right and what to do after.
If the slope of the tangent line is 0, its normal will be a vertical line of the form x = k, where k is the x-value at the point of tangency.
fulton33 said:
12. I am lost here as well. The tangent slope would be DNE, which would mean the normal slope to be 0. When I plug 0 in for m and (2,0) for x and y I get y=0. I think that is wrong.
This is the converse of #10. If the tangent slope is undefined, a line perpendicular to it will have slope 0.
 
  • #3
Mark44 said:
If the slope of the tangent line is 0, its normal will be a vertical line of the form x = k, where k is the x-value at the point of tangency.
This is the converse of #10. If the tangent slope is undefined, a line perpendicular to it will have slope 0.

10. Does that mean for number 10 the equation would be x=0 because at x=-1 f'(-1)=0?

12. Does that mean that for number 12 y=0 would then be correct?
 
  • #4
fulton33 said:
Does that mean for number 10 the equation would be x=0 because at x=-1 f'(-1)=0?
No. You're given information about the point (-1, -2).
fulton33 said:
Does that mean that for number 12 y=0 would then be correct?
Yes.
 
  • #5
Mark44 said:
No. You're given information about the point (-1, -2).
Yes.
That makes sense about 10. That would make it x=-1?
 
  • #6
Yes, that's better.
 

1. How do you calculate the slope of a tangent line?

To calculate the slope of a tangent line at a given point on a curve, you can use the derivative of the function at that point. The derivative represents the rate of change of the function at that point and can be found using the limit definition of derivative or by using differentiation rules.

2. What is the equation for a tangent line?

The equation for a tangent line at a given point on a curve is y = mx + b, where m represents the slope of the tangent line and b represents the y-intercept. To find the equation, you can use the point-slope form of a line and plug in the coordinates of the given point.

3. How do you find the equation for a normal line?

The equation for a normal line at a given point on a curve is y = (-1/m)x + b, where m represents the slope of the tangent line and b represents the y-intercept. To find the equation, you can first find the slope of the tangent line using the derivative and then take the negative reciprocal to find the slope of the normal line.

4. Can you use the tangent and normal line equations to find the equation of a curve?

No, the tangent and normal line equations only provide information about the slope of the curve at a given point. To find the equation of a curve, you would need more information such as multiple points on the curve or the general form of the equation.

5. How do you use tangent and normal lines to approximate a curve?

Tangent and normal lines can be used to approximate a curve by finding the equation of the line at a given point and using it to estimate the value of the function. As the distance between the points on the curve becomes smaller, the approximation becomes more accurate.

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