Math Problem Statement

Find the equation of the line tangent to the graph of the function w(x) = -10/(x^2 + 4) at the point (1, 2).

Solution

To solve the problem shown in the image, we need to find the equation of the tangent line to the curve w(x)=10x2+4w(x) = \frac{-10}{x^2 + 4} at the point (1,2)(1, 2).

Steps:

  1. Find the derivative w(x)w'(x): To get the slope of the tangent line, we need the derivative of w(x)w(x).

    The function is: w(x)=10x2+4w(x) = \frac{-10}{x^2 + 4}

    Using the quotient rule: w(x)=(x2+4)(0)(10)(2x)(x2+4)2=20x(x2+4)2w'(x) = \frac{(x^2 + 4)(0) - (-10)(2x)}{(x^2 + 4)^2} = \frac{20x}{(x^2 + 4)^2}

  2. Evaluate the derivative at x=1x = 1 to find the slope: Substituting x=1x = 1 into the derivative:

    w(1)=20(1)(12+4)2=2025=0.8w'(1) = \frac{20(1)}{(1^2 + 4)^2} = \frac{20}{25} = 0.8

    So, the slope of the tangent line at (1,2)(1, 2) is 0.8.

  3. Use the point-slope form of the equation of a line: The point-slope form is given by: yy1=m(xx1)y - y_1 = m(x - x_1) where mm is the slope and (x1,y1)(x_1, y_1) is the point of tangency.

    Plugging in m=0.8m = 0.8, x1=1x_1 = 1, and y1=2y_1 = 2: y2=0.8(x1)y - 2 = 0.8(x - 1) Simplifying: y=0.8x+1.2y = 0.8x + 1.2

Thus, the equation of the tangent line is: y=0.8x+1.2y = 0.8x + 1.2

Let me know if you want further details, or if you have any questions!

Related Questions:

  1. What is the general formula for the quotient rule in calculus?
  2. How do you find the second derivative of the function w(x)w(x)?
  3. What happens to the tangent line if the point of tangency changes?
  4. How can you check if a line is tangent to a curve using limits?
  5. What is the geometric interpretation of the slope of a tangent line?

Tip:

When using the quotient rule, make sure to correctly apply the formula f(x)g(x)f(x)g(x)g(x)2\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} to avoid calculation errors!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Tangent Lines
Quotient Rule

Formulas

Quotient Rule: (f'(x)g(x) - f(x)g'(x)) / g(x)^2
Point-Slope Form: y - y1 = m(x - x1)

Theorems

Quotient Rule
Tangent Line Theorem

Suitable Grade Level

Grades 11-12 or Introductory College Level