Find the Tangent lines of the slopes of the three zeroes

JamesonIn summary, the conversation discusses finding the derivative and slopes of tangent lines for the equation f(x)=1+\frac{50 \sin(x)}{x^2+3}, with the additional information that the equation has three zeroes at 0.02, 3.16, and -3.12. The conversation also includes a hint to use the quotient rule for finding the derivative.
  • #1
Nivetham
2
0
1+50sinx/x^2+3
-5 < x < 5

3 zeroes: 0.02, 3.16, -3.12
Find the derivative and the slopes of the tangent lines.

I need help with the last part. I found out the three zeroes by adding and subtracting pi from the equation at top by setting it to zero.
Thank you!
 
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  • #2
Hi Nivetham,

Welcome to MHB! :)

Is this your equation? \(\displaystyle f(x)=1+\frac{50 \sin(x)}{x^2}+3?\) or is it \(\displaystyle f(x)=1+\frac{50 \sin(x)}{x^2+3}\), or is it \(\displaystyle \frac{1+50 \sin(x)}{x^2+1}\)?

Jameson
 
  • #3
Hi! It's the second equation. Thank you!
 
  • #4
Nivetham said:
Hi! It's the second equation. Thank you!
Hello Nivetham,
Have you derivate the function?
Hint: Quotient rule

Regards,
 

FAQ: Find the Tangent lines of the slopes of the three zeroes

What is the significance of finding the tangent lines of the slopes of the three zeroes?

Finding the tangent lines of the slopes of the three zeroes is important in calculus as it allows us to determine the rate of change at those points. This is useful in many real-world applications, such as predicting the growth of a population or the speed of an object at a specific time.

How do you find the tangent lines of the slopes of the three zeroes?

To find the tangent lines, we first need to find the derivative of the function at the points where the zeroes occur. Then, we can plug these values into the equation of a line, y = mx + b, where m is the slope and b is the y-intercept. This will give us the equation of the tangent line at each zero.

Can there be more than one tangent line at a zero?

Yes, there can be more than one tangent line at a zero. This occurs when the function has a horizontal tangent at that point, meaning the slope of the function is zero. In this case, there are infinite tangent lines that can be drawn at that zero.

What information can we gather from the tangent lines of the slopes of the three zeroes?

By finding the tangent lines, we can determine the instantaneous rate of change at each zero. We can also use the slope of the tangent line to determine whether the function is increasing or decreasing at that point.

Are there any limitations to finding the tangent lines of the slopes of the three zeroes?

One limitation is that the function must be differentiable at the points where the zeroes occur. This means that the function must be continuous and smooth at those points. Additionally, finding the tangent lines only gives us information about the function at those specific points, and we cannot make assumptions about the rest of the function based on these lines alone.

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