Problem 67 - Calculus Single Variable

Question

Find all values of \(c\) for which the tangent lines to the graphs of \(f(x)=x^{3}-8 x+3\) and \(g(x)=4 / x\) at \((c, f(c))\) and (c, \(g(c)\) ) are parallel.

Step-by-Step Solution

Answer

Parallel tangents occur at \(c = \pm \sqrt{2}\) and \(c = \pm \sqrt{\frac{2}{3}}\).

Step 1: Find the derivative of the functions

To find the tangent line, we first need the slopes of the tangent lines at any given point. We compute the derivatives:For function \(f(x) = x^3 - 8x + 3\):\[ f'(x) = 3x^2 - 8 \]For function \(g(x) = \frac{4}{x}\):\[ g'(x) = -\frac{4}{x^2} \]

Step 2: Set the derivatives equal for parallel lines

For the tangent lines to be parallel at \(x = c\), the slopes must be equal. Therefore, set the derivatives equal:\[ 3c^2 - 8 = -\frac{4}{c^2} \]

Step 3: Solve the equation for c

Multiply both sides by \(c^2\) to eliminate the fraction and simplify:\[ c^2(3c^2 - 8) = -4 \]\[ 3c^4 - 8c^2 + 4 = 0 \]Introduce a substitution, let \(u = c^2\), thus:\[ 3u^2 - 8u + 4 = 0 \]

Step 4: Solve the quadratic equation

Solve \(3u^2 - 8u + 4 = 0\) using the quadratic formula:\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]with \(a = 3\), \(b = -8\), and \(c = 4\):\[ u = \frac{8 \pm \sqrt{64 - 48}}{6} \]\[ u = \frac{8 \pm \sqrt{16}}{6} \]\[ u = \frac{8 \pm 4}{6} \]\[ u = 2 \text{ or } u = \frac{2}{3} \]

Step 5: Determine the values for c

Recall \(u = c^2\), so we need \(c\):For \(u = 2\):\[ c^2 = 2 \Rightarrow c = \pm \sqrt{2} \]For \(u = \frac{2}{3}\):\[ c^2 = \frac{2}{3} \Rightarrow c = \pm \sqrt{\frac{2}{3}} \]

Step 6: Conclusion

Therefore, the values of \(c\) for which the tangent lines are parallel are \(c = \pm \sqrt{2}\) and \(c = \pm \sqrt{\frac{2}{3}}\).

Key Concepts

Tangent Lines

Tangent lines are lines that touch a curve at a single point. This point of tangency is important because the tangent line at that point represents the instantaneous rate of change of the function, providing a linear approximation of the curve. Think of it as the "best linear fit" at a specific point on the curve. Calculating the tangent line at a certain point involves using derivatives, as the slope of the tangent line is the same as the derivative of the function at that point.

• Tangent lines only touch the curve at one precise point without cutting across it.
• Their slopes reflect how steep the curve is at the point of tangency.
• In exercise problems, calculating tangent lines often requires finding derivatives first.

Understanding tangent lines is crucial, especially in calculus problems where they are used to approximate values and solve for points of intersection or parallelism.

Derivatives

Derivatives are a cornerstone of calculus, representing the rate at which a function is changing at any given point. They allow us to understand how a function's output value changes as its input value changes, essentially giving the slope of a function's graph at any given point. For the function provided, the derivatives were calculated as follows:

For the function \( f(x) = x^3 - 8x + 3 \), the derivative is:
\[ f'(x) = 3x^2 - 8 \] For the function \( g(x) = \frac{4}{x} \), the derivative is:
\[ g'(x) = -\frac{4}{x^2} \]

• Derivatives are used to determine the slope of the tangent line to the graph of a function at any given point.
• They provide critical information about the behavior of functions, such as maxima, minima, and points of inflection.

By mastering derivatives, students can accurately analyze and predict the behavior of different functions.

Quadratic Equation

Quadratic equations appear frequently in mathematics, characterized by their standard form \( ax^2 + bx + c = 0 \). They are used to find the roots, which are the values of the variable that satisfy the equation. In many cases, such as in this exercise, the quadratic formula is employed to solve these equations:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In the given solution:

• The equation was transformed into quadratic form after algebraic manipulation: \( 3u^2 - 8u + 4 = 0 \).
• The quadratic formula was used to find the possible values of \( u \) (an auxiliary variable introduced for ease), resulting in two potential solutions: \( u = 2 \) and \( u = \frac{2}{3} \).

Quadratic equations offer effective ways to solve many real-world and theoretical problems, making them a vital part of mathematics.

Parallel Lines

In geometry, parallel lines are lines in a plane that never meet. They are always the same distance apart and have the same slope. Therefore, for two lines to be parallel, they must have equal slopes. In calculus problems, setting the derivatives (which represent slopes) equal to each other at a point determines if two tangent lines are parallel.

In the exercise, by setting the derivatives equal, we find the values of \( c \) for which the tangent lines to \( f(x) \) and \( g(x) \) are parallel:

• The condition of parallelism led to the equation \( 3c^2 - 8 = -\frac{4}{c^2} \).
• This equation was then solved to find specific values for \( c \) ensuring that the tangent lines at those points have the same slope.

Recognizing and solving for parallel lines help with identifying and solving intersection and optimization problems in calculus.