Math Problem Statement

Find the equations of tangent lines and points on graphs where tangents have specific slopes or are horizontal.

Solution

Let's solve the problems displayed in the image one by one.


Problem 13:

Find an equation of the tangent line to the graph of f(x)f(x) at the specified point (1,3)(1, 3).

Given function: f(x)=(x3+2)(3x24x+2)f(x) = (x^3 + 2)(3x^2 - 4x + 2)

  1. First, differentiate f(x)f(x) to find f(x)f'(x), which represents the slope of the tangent line at any point.
  2. Evaluate f(x)f'(x) at x=1x = 1 to find the slope of the tangent line at (1,3)(1, 3).
  3. Use the point-slope form to find the equation of the tangent line: yy1=m(xx1)y - y_1 = m(x - x_1) where (x1,y1)=(1,3)(x_1, y_1) = (1, 3) and m=f(1)m = f'(1).

Problem 14:

Find an equation of the tangent line to the graph of the following function ff at the specified point (3,3)(3, 3).

Given function: f(x)=8xx21f(x) = \frac{8x}{x^2 - 1}

  1. Differentiate f(x)f(x) to find f(x)f'(x).
  2. Evaluate f(x)f'(x) at x=3x = 3 to get the slope of the tangent line at that point.
  3. Use the point-slope form to find the equation of the tangent line, with the point (3,3)(3, 3) and the calculated slope.

Problem 15:

Find the point(s) on the graph of ff where the tangent line is horizontal.

Given function: f(x)=xx2+81f(x) = \frac{x}{x^2 + 81}

  1. A horizontal tangent line means f(x)=0f'(x) = 0.
  2. Differentiate f(x)f(x) and set f(x)=0f'(x) = 0 to find the critical points.
  3. Solve for the xx-values where this occurs.
  4. Evaluate f(x)f(x) at these xx-values to get the corresponding yy-values, giving you the points where the tangent is horizontal.

Problem 16:

Find the point(s) on the graph of the following function where the slope of the tangent line is equal to 92-\frac{9}{2}.

Given function: f(x)=x+9x29f(x) = \frac{x + 9}{x^2 - 9}

  1. Differentiate f(x)f(x) to find f(x)f'(x).
  2. Set f(x)=92f'(x) = -\frac{9}{2} and solve for xx.
  3. Use the solved xx-values to find the corresponding yy-values by plugging them into f(x)f(x).

Would you like the full solutions for these problems? Here are five follow-up questions to think about:

  1. What is the chain rule and how is it applied in differentiating the first function?
  2. How does one differentiate a quotient of two functions, as seen in Problem 14?
  3. What conditions are necessary for a tangent line to be horizontal?
  4. How do you solve for critical points where the slope is zero or equal to a specific value?
  5. What role does the second derivative play in analyzing the curvature of functions at these critical points?

Tip: Always check if a function is differentiable at the given point before attempting to find the tangent line.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Tangent Lines
Slopes
Critical Points

Formulas

Derivative formulas
Point-slope form of a line
Quotient rule
Product rule

Theorems

Differentiability
Tangent line approximation

Suitable Grade Level

College