Problem 82 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line has slope 1 . $$ y=-0.025 x^{2}+4 x $$

Step-by-Step Solution

Answer

The point is \( (60, 150) \).

Step 1: Find the derivative

The derivative of a function gives us the slope of the tangent line at any point on the graph. To find the derivative of the function \( y = -0.025x^2 + 4x \), differentiate each term individually.

Step 2: Differentiate the function

Differentiate \( -0.025x^2 \) to get \( -0.05x \) and differentiate \( 4x \) to get 4. Therefore, the derivative \( \frac{dy}{dx} \) or \( y' = -0.05x + 4 \).

Step 3: Set the derivative equal to the desired slope

We are looking for points where the slope of the tangent is 1. Set the derivative equal to 1: \( -0.05x + 4 = 1 \).

Step 4: Solve for \( x \)

Subtract 4 from both sides: \( -0.05x = -3 \). Then, divide by \(-0.05\) to solve for \( x \): \( x = 60 \).

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

Substitute \( x = 60 \) back into the original function to find the \( y \)-coordinate: \( y = -0.025(60)^2 + 4(60) \).

Step 6: Calculate \( y \, \text{} "coordinate" \)

Calculate \( 60^2 = 3600 \). Compute \( y = -0.025(3600) + 240 = -90 + 240 = 150 \).

Step 7: Point on the graph

The point on the graph where the tangent line has a slope of 1 is \( (60, 150) \).

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at just one point. This point is special because the tangent line has the same slope as the curve at that point. When we talk about the slope of the tangent line, we're referring to how steep the line is.

• Tangent lines are important because they show the direction a curve is going at a specific point.
• They help us understand the behavior of curves at intricate parts.
• In calculus, tangent lines give insights into instantaneous rates of change.

While a tangent line only touches the curve at one point, it gives a perfect snapshot of the curve's trend right there. If you imagine a car on a curved road, the tangent line would indicate the car's direction at the exact moment it passes the curve.

Derivative

In calculus, a derivative is a way to show how a function changes as its input changes. It's a core tool to find the slope of a curve at any given point.

• The derivative of a function at a particular point is the slope of the tangent line to the graph of the function at that point.
• Finding the derivative involves a process called differentiation.

Differentiation tells you how to compute this slope. For example, if you have a function like \( y = -0.025x^2 + 4x \), differentiation helps us find how quickly "y" is changing for any "x".Differentiation rules:

• The power rule allows us to differentiate terms like \( x^n \), and multiply the result by its exponent.
• For \( y = -0.025x^2 + 4x \), this gives us a derivative of \( y' = -0.05x + 4 \).

In real terms, derivatives let us predict trends, speeds, and changes, which is why they are fundamental in sciences, engineering, and economics.

Slope of Tangent

The slope of a tangent line is a measure of how steep that line is at a specific point on a curve. Calculus allows us to calculate this using derivatives.

• The slope of the tangent is given by the value of the derivative at the point of interest.
• If the slope is zero, the tangent line is horizontal, which implies the function might have a maximum or minimum point there.

To find the slope of a tangent:

• First, calculate the derivative of the function representing the curve.
• Evaluate this derivative at the specific point to get the slope.

In the exercise, we were interested in finding where the tangent line has a slope of 1. By setting the derivative equal to 1 and solving for "x", we discovered the point (60, 150). This means at point (60, 150), the curve's slope and the tangent line's slope are both exactly 1, illustrating an upward but gentle incline at that position.