Problem 38 - Thomas Calculus

Question

Find the point on the graph of \(y=20 x^{3}+60 x-3 x^{5}-5 x^{4}\) with the largest slope.

Step-by-Step Solution

Answer

The point with the largest slope is (0, 0).

Step 1: Understand the Function

We are given the function \( y = 20x^3 + 60x - 3x^5 - 5x^4 \). We need to find the point on this graph where the slope of the tangent is the largest. This means we need to find the maximum value of the derivative.

Step 2: Find the Derivative

The slope of the tangent line at any point on the graph of a function is given by its derivative. So, we first differentiate the function: \[ y' = \frac{d}{dx}(20x^3 + 60x - 3x^5 - 5x^4) \]Using the power rule, we get:\[ y' = 60x^2 + 60 - 15x^4 - 20x^3 \].

Step 3: Find Critical Points of the Derivative

To find the maximum slope, we need to find the critical points of the derivative \( y' \). These are where \( y'' = 0 \) or \( y'' \) is undefined. Differentiate again:\[ y'' = \frac{d}{dx}(60x^2 + 60 - 15x^4 - 20x^3) \]\[ y'' = 120x - 60x^3 - 60x^2 \].Set \( y'' = 0 \):\[ 120x - 60x^3 - 60x^2 = 0 \].

Step 4: Solve for Critical Points

Factor the equation \( 120x - 60x^3 - 60x^2 = 0 \):\[ 60x(2 - x^2 - x) = 0 \].Solve for x: \[ x = 0 \] or solve \( 2 - x^2 - x = 0 \). The quadratic equation can be solved using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = -1, b = -1, c = 2 \). This gives us roots \( x \approx 1.28078 \) and \( x \approx -1.28078 \).

Step 5: Evaluate Derivative at Critical Points

Substitute the critical points into the first derivative to find the slopes: \[ y'(0) = 60 \] \[ y'(1.28078) \lessapprox 0 \] \[ y'(-1.28078) \lessapprox 0 \] The largest value of \( y' \) occurs at \( x = 0 \).

Step 6: Find the Function Value at this Point

Plug \( x = 0 \) back into the original equation to find the y-coordinate of the point:\[ y = 20(0)^3 + 60(0) - 3(0)^5 - 5(0)^4 = 0 \]. Thus, the point on the graph is \((0, 0)\).

Key Concepts

Critical Points

Critical points are essential in analyzing where functions reach maximum or minimum values. For a function, these are points where its derivative, typically denoted as \( y' \), is either zero or undefined. At these points, the function might switch direction, portraying a peak (maximum), a dip (minimum), or a flat stretch (saddle point). To determine critical points, follow these steps:

• Firstly, calculate the derivative of the function. This represents the slope of the tangent line at any point on the function's curve.
• Set this derivative equal to zero (\( y' = 0 \)) or find where it is undefined to identify all potential candidates for critical points.

Determining these points can help identify where the function behaves in a structured way, especially in applications like physics for determining equilibrium points in systems.

Power Rule

The power rule is a simple yet pivotal tool in calculus for differentiating functions with exponents. This rule states that if you have a function \( f(x) = x^n \), then its derivative is \( f'(x) = nx^{n-1} \). It simplifies the process of differentiation as follows:

• Take the power \( n \) of the variable \( x \).
• Multiply the entire term by this power.
• Decrease the power of \( x \) by 1.

For example, applying the power rule to differentiate \( 20x^3 \) results in \( 60x^2 \). This method is straightforward, making it a fundamental part of any calculus toolkit when dealing with polynomial functions.

Tangent Line Slope

The slope of the tangent line to a curve at a given point provides critical insight into the curve's direction and steepness at that precise moment. It is found by taking the derivative of the function, \( y'(x) \). Here's what to consider:

• The derivative at any point \( x \) gives the slope of the tangent line to the curve there.
• A positive slope indicates the function is increasing at that point, while a negative slope shows it is decreasing.
• A slope of zero means the tangent line is horizontal, often indicating a local maximum or minimum at the point.

In the example, we aimed to find where the tangent line's slope is the largest by determining the maximum value of the derivative, guiding us to the point \((0, 0)\) as the location of the largest slope.

Quadratic Formula

The quadratic formula is crucial for solving quadratic equations, which appear in many calculus problems once derivatives have been set to zero. For an equation in the standard form \( ax^2 + bx + c = 0 \), it is solved using:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here’s how it breaks down:

• Calculate the discriminant \( b^2 - 4ac \). If this is positive, you get two real roots. If zero, there is one real root; if negative, the roots are complex.
• Plug these values into the formula to find the values of \( x \) that satisfy the equation.

For our problem, using the quadratic formula helped find the potential critical points \( x \approx 1.28078 \) and \( x \approx -1.28078 \), aiding in understanding where the slope might reach its maximum or minimum values.