Question

Complete the following statement: If \(\frac{d y}{d x}\) is small, then small changes in \(x\) result in relatively _________ changes in the value of \(y\).

Short answer

Question: Fill in the blank in the following statement: If the derivative of a function (dy/dx) is small, then small changes in x result in _____ changes in the value of y. Answer: relatively small

Step-by-step solution

Understand the relationship between the derivative of a function and the changes in x and y

The derivative of a function at a specific point is a measure of how much the function's output (y) will change with respect to small changes in the input (x). In simple terms, the derivative tells us the slope of the tangent line to the function at that point. When the derivative value is small, it means that the slope of the tangent line is close to zero or small. Therefore, the function is not changing much with respect to its input. In this condition, small changes in input will result in relatively small changes in the output.

Complete the statement

Based on our understanding from step 1, we can now confidently fill in the blank to complete the statement. If \(\frac{d y}{d x}\) is small, then small changes in \(x\) result in relatively small changes in the value of \(y\).

Key concepts

Rate of Change

Imagine you are watching a car climb a steep hill. The faster it goes up, the steeper and more noticeable the climb is. Similarly, in calculus, the concept of "rate of change" revolves around how fast one quantity changes in relation to another. The derivative, denoted as \( \frac{d y}{d x} \), is essentially the mathematical tool that helps measure this rate of change. Think of it as a speedometer for functions.

• Positive rate of change: When \( \frac{d y}{d x} \) is positive, \( y \) increases as \( x \) increases. The function moves upwards.
• Negative rate of change: When \( \frac{d y}{d x} \) is negative, \( y \) decreases as \( x \) increases. The function moves downwards.
• Zero rate of change: When \( \frac{d y}{d x} \) equals zero, \( y \) doesn't change as \( x \) changes, indicating a flat line or a point of extremity such as a peak or trough.

Understanding the rate of change is crucial for predicting how slight adjustments in one variable may impact another, giving us insights into the dynamics of various phenomena.

Slope of Tangent Line

Think of the slope of a tangent line as a snapshot of how steep a curve is at a particular point. When you have a graph of a function, a tangent line will touch the curve at just one point without crossing it. The slope of this line represents the derivative at that point, showing the curve's immediate direction.

• Small slope: If \( \frac{d y}{d x} \) is small, the tangent line is almost flat or slightly sloped. This means the function is changing gradually at that point.
• Large slope: A large \( \frac{d y}{d x} \) indicates a steep tangent, revealing significant changes in the function with small changes in \( x \).

Understanding the slope of a tangent line helps in visualizing and predicting the behavior of a function at any given point on its graph. It's like understanding the gradient of a hill—all from a single spot on your journey.

Function Behavior

Let’s think about functions like a story unfolding. The function behavior tells us how the story develops based on different inputs and how it reacts.

• Increasing behavior: When the derivative is positive, the function tells a tale of growth or increase.
• Decreasing behavior: A negative derivative implies the function is in a downward trend.
• Static behavior: Zero derivative means the function is in a pause, neither going up nor down. This may suggest an important point such as a maximum, minimum, or point of inflection.

Understanding function behavior is key to examining how a function evolves. It’s like getting to know the rhythm and tempo of a story that the function narrates, how it reaches its highs and lows.