Problem 67 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=-0.01 x^{2}+0.4 x+50 $$

Step-by-Step Solution

Answer

The tangent line is horizontal at (20, 54).

Step 1: Identify the condition for a horizontal tangent line

A tangent line is horizontal when its slope is equal to zero. The slope of the tangent line is given by the derivative of the function.

Step 2: Differentiate the function

Differentiate the function \[ y = -0.01x^2 + 0.4x + 50 \]using power rule for differentiation to find the derivative:\[ y' = rac{d}{dx}(-0.01x^2 + 0.4x + 50) = -0.02x + 0.4. \]

Step 3: Set the derivative equal to zero for horizontal tangents

Set the derivative equal to zero to find the values of \( x \) where the tangent line is horizontal:\[ -0.02x + 0.4 = 0. \]

Step 4: Solve the equation for \( x \)

Solve the equation \( -0.02x + 0.4 = 0 \) to find the point where the tangent is horizontal:\[ -0.02x = -0.4 \]\[ x = rac{-0.4}{-0.02} = 20. \]

Step 5: Find the \( y \)-coordinate

Substitute \( x = 20 \) back into the original function to find the \( y \)-coordinate:\[ y = -0.01(20)^2 + 0.4(20) + 50 \]\[ y = -0.01(400) + 8 + 50 \]\[ y = -4 + 8 + 50 = 54. \]

Step 6: State the point of the horizontal tangent

The tangent line is horizontal at the point \((20, 54)\).

Key Concepts

Derivatives in Calculus

Derivatives are foundational in calculus and help describe how a function changes at any point. A derivative is essentially a rate of change. You can think of it as answering the question, "How fast is this function increasing or decreasing?" For a graph, the derivative at a specific point gives the slope of the tangent line to the curve at that point.

When we talk about tangent lines, these are straight lines that touch a curve at a single point without crossing it. At this point, the slope of the tangent line is the same as the slope of the curve itself. To find this slope, we use derivatives. This is why they are sometimes called the 'instantaneous rate of change.'

In simpler terms, if we have a function describing a car's journey, taking the derivative gives us the car's speed at any instant on that journey. Understanding derivatives is crucial not only for solving problems like finding horizontal tangents but also for many other applications, from physics to economics.

Power Rule Differentiation

The power rule is one of the most straightforward and common rules for differentiation used in calculus. It is specifically useful when differentiating polynomial functions. The power rule states that if you have a function in the form of \( y = ax^n \), then the derivative \( y' \) is given by \( y' = anx^{n-1} \).

Let's break this down with our example that contains the function \( y = -0.01x^2 + 0.4x + 50 \):

• For the term \(-0.01x^2\), the derivative would be \(-0.02x\). Here, we multiplied \(-0.01\) by \(2\) and subtracted \(1\) from the power to make it linear.
• The term \(0.4x\) differentiates to \(0.4\) since the power of \(x\) is \(1\), which reduces it to a constant when derivative is taken.
• The constant \(50\) disappears, because the derivative of a constant is \(0\).

Together, this gives us the derivative \( y' = -0.02x + 0.4 \). This derivative can now be used to find out when and where a function's slope, or the tangent line, is horizontal.

Slope of Tangent Line

The slope of a tangent line is vital when analyzing how a function behaves at a particular point on its graph. When the slope is zero, it means that the tangent line is perfectly horizontal—a crucial scenario in many optimization and analytical problems.

To find a horizontal tangent line, you need to look at the derivative of the function. As determined in our previous work, the derivative of the function \( y = -0.01x^2 + 0.4x + 50 \) is \( y' = -0.02x + 0.4 \). Set this derivative equal to zero to find when the slope becomes horizontal: \(-0.02x + 0.4 = 0\).

Solving this equation, we find \( x = 20 \). This tells us that at \( x = 20 \), the slope of the tangent line is zero. To fully identify the point, substitute \( x = 20 \) back into the original function to find its \( y \)-coordinate, which gives \( y = 54 \). Thus, the slope of the tangent line is horizontal at the point \((20, 54)\).

Understanding how to find and interpret the slope of tangent lines facilitates deeper insights into the behavior of functions across various fields.