Problem 66 - Calculus Single Variable

Question

Find all values of \(c\) for which the tangent lines to the graphs of \(f(x)=x^{2}-7 x+9\) and \(g(x)=9 / x\) at \((c, f(c))\) and \((c, g(c))\) are parallel.

Step-by-Step Solution

Answer

The value of \(c\) for which the tangent lines are parallel is \(c = 3\).

Step 1: Find the derivatives for both functions

To find where the tangent lines are parallel, we need the derivatives of both functions, because parallel lines have equal slopes. For the function \(f(x) = x^2 - 7x + 9\), the derivative is \(f'(x) = 2x - 7\). For the function \(g(x) = \frac{9}{x}\), the derivative using the power rule is \(g'(x) = -\frac{9}{x^2}\).

Step 2: Set the derivatives equal to find the value of c

To find the value(s) of \(c\) for which the tangent lines are parallel, set \(f'(c) = g'(c)\). This gives us the equation \(2c - 7 = -\frac{9}{c^2}\).

Step 3: Solve for c

First, multiply both sides by \(c^2\) to get rid of the fraction, resulting in \(c^2(2c - 7) = -9\). This simplifies to \(2c^3 - 7c^2 + 9 = 0\).

Step 4: Use the rational root theorem or factoring

To solve the cubic equation \(2c^3 - 7c^2 + 9 = 0\), try using a method such as factoring, synthetic division, or the rational root theorem to find possible rational roots. The rational root theorem suggests possible roots of \(\pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}\).

Step 5: Test possible rational roots

Try potential roots by substitution into the equation to see which make it zero. After testing various possibilities, \(c = 3\) works, as substituting into the equation gives \(2(3)^3 - 7(3)^2 + 9 = 54 - 63 + 9 = 0\). Feel free to verify by substitution, or use numerical methods or graphing for further validation.

Step 6: Verify parallel condition with calculated c

Verify that when \(c = 3\), the derivatives (slopes) are equal. Substituting \(c = 3\) back into the derivatives: \(f'(3) = 2(3) - 7 = -1\) and \(g'(3) = -\frac{9}{3^2} = -1\), showing the slopes are indeed equal and the tangent lines are parallel.

Key Concepts

Derivative

The derivative of a function provides a way to measure how a function's output changes as its input changes—essentially it gives us the slope of the tangent line at any given point on the function.

For a function such as a polynomial, we can find the derivative using straightforward calculus rules. In the example of a quadratic function like \( f(x) = x^2 - 7x + 9 \), the differentiation rule for power functions can be used, resulting in its derivative \( f'(x) = 2x - 7 \).

For rational functions like \( g(x) = \frac{9}{x} \), derivatives are computed differently. Using the power rule, the function can be rewritten as \( 9x^{-1} \), and its derivative becomes \( g'(x) = -\frac{9}{x^2} \).

Derivatives are crucial when solving problems of parallel tangent lines, as it is the equality of these derivatives that determines when line slopes match.

Slope

Slope is a measure of the steepness of a line, often denoted by the letter \( m \). It describes the rate of change between two variables, typically \( x \) and \( y \), and is defined mathematically as the ratio of the rise over the run between two points on a line.

In the context of tangent lines, the slope at a single point is represented by the derivative at that point.

• If two tangent lines are parallel, they share the same slope.
• This means their derivatives (slopes) at certain points are equal.

For example, in the functions \( f(x) \) and \( g(x) \) given, to find when their tangent lines are parallel, the derivatives \( f'(x) \) and \( g'(x) \) must be equal.

Cubic Equation

A cubic equation is an algebraic equation of the form \( ax^3 + bx^2 + cx + d = 0 \), where the highest degree is 3. Solving cubic equations can be more complex than solving quadratics due to the extra degree, which means potentially more solutions.

In this exercise, after equating derivatives \( f'(c) = g'(c) \) to find when tangent lines are parallel, we arrive at a cubic equation \( 2c^3 - 7c^2 + 9 = 0 \).

Cubic equations can be tackled using several methods including:

• Factoring, if possible, when obvious factors are seen.
• The rational root theorem to suggest possible roots.
• Nesting methods like synthetic division to systematically solve.

Testing values can then help identify roots that satisfy the equation.

Rational Root Theorem

The Rational Root Theorem is a principle used to find potential rational solutions of polynomial equations. It asserts that any rational solution, or root, of the polynomial equation \( ax^n + bx^{n-1} + Cd = 0 \) must be a fraction \( \frac{p}{q} \), where \( p \) divides the constant term \( d \) and \( q \) divides the leading coefficient \( a \).

In the example \( 2c^3 - 7c^2 + 9 = 0 \), the Rational Root Theorem helps by pinpointing possible rational roots:

• \( \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \)
• By systematically testing these values in the equation, \( c = 3 \) emerges as a valid solution.

Using the Rational Root Theorem effectively narrows down candidates, making the process of identifying true solutions much more manageable.