Problem 39 - Single Variable Calculus

Question

Find the point \((s)\) on the graph of the function at which the tangent line has the indicated slope. \(f(x)=2 x^{3}+3 x^{2}-12 x-10 ; m_{\text {tan }}=0\)

Step-by-Step Solution

Answer

The points \((s)\) on the graph of the function \(f(x) = 2x^3 + 3x^2 - 12x - 10\) with a tangent line slope of 0 are \((-2, 16)\) and \((1, -17)\).

Step 1: Find the derivative of the function\(f(x)\)

To find the derivative of the function \(f(x)\), apply the power rule to each term: \(f'(x) = \frac{d}{dx}(2x^3) + \frac{d}{dx}(3x^2) - \frac{d}{dx}(12x) - \frac{d}{dx}(10)\) \(f'(x) = 6x^2 + 6x - 12\)

Step 2: Set the derivative equal to 0

As we want to find the point where the tangent line has a slope equal to 0, we set \(f'(x) = 0\): \(6x^2 + 6x - 12 = 0\)

Step 3: Solve the equation for x

Divide the equation by 6 to simplify: \(x^2 + x - 2 = 0\) We can now factor the quadratic expression: \((x + 2)(x - 1) = 0\) Solve for x: \(x = -2\) or \(x = 1\)

Step 4: Find the coordinates \((s)\)

Plug the x-values found in step 3 back into the original function to find the corresponding y-values: For \(x = -2\): \(f(-2) = 2(-2)^3 + 3(-2)^2 - 12(-2) - 10 = 16\) So, point \((-2, 16)\) has a tangent with a slope of 0. For \(x = 1\): \(f(1) = 2(1)^3 + 3(1)^2 - 12(1) - 10 = -17\) So, point \((1, -17)\) has a tangent with a slope of 0. So, the points with a tangent line slope of 0 are \((-2, 16)\) and \((1, -17)\).

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it.
At that point, the slope of the tangent line gives us valuable information about the behavior of the function.
In this exercise, we are interested in finding the points where the slope of the tangent line is zero because this indicates the function is neither increasing nor decreasing at that point.

• A slope of zero suggests a horizontal tangent line.
• To find the tangent line, we first calculate the derivative of the function.
• We then solve for when the derivative equals zero to find potential points of zero slope.

Power Rule

The power rule is a fundamental tool in calculus used to take derivatives of polynomial functions quickly.
It states if you have a term in the form of \(ax^n\), its derivative will be \(anx^{n-1}\). This allows us to find the rate of change of each term individually.

• For the term \(2x^3\), the derivative is \(6x^2\).
• For \(3x^2\), it simplifies to \(6x\).
• \(-12x\) becomes \(-12\).
• Constants like \(-10\) have a derivative of zero as they do not change.

By applying the power rule, we can find the derivative of the entire function, simplifying and solving it for different values of \(x\).

Quadratic Factoring

Quadratic factoring is a process used to solve quadratic equations, helping us find the roots or solutions of equations like \(x^2 + x - 2 = 0\).
This involves expressing the quadratic equation in terms of its factors.

• Think of the original equation as the product of two simpler expressions.
• For \(x^2 + x - 2 = 0\), it factors to \((x + 2)(x - 1) = 0\).
• Setting each factor equal to zero gives the solution \(x = -2\) or \(x = 1\).

This method is efficient and provides quick answers, essential for finding where the derivative equals zero.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of a variable. They can represent a wide array of real-world scenarios and are often graphed as smooth, curved lines.

• In the example \(f(x) = 2x^3 + 3x^2 - 12x - 10\), it's a cubic polynomial, featuring powers up to \(x^3\).
• Polynomials are continuous and differentiable, making them suitable candidates for analysis using calculus.
• Understanding their shape and behavior involves studying their derivatives and roots.

By examining the critical points where the derivative is zero, we gain insights into the turning points, helping us understand how the function changes direction at different intervals.