Question

Using the Alternative Form of the Derivative In Exercises \(65-74,\) use the alternative form of the derivative to find the derivative at \(x=c\) (if it exists). $$ f(x)=3 / x, \quad c=4 $$

Short answer

The derivative of the function \(f(x)=3/x\) at \(x=4\) using the alternative form of the derivative is \(-3/16.\)

Step-by-step solution

Define the Alternate form of the Derivative

The alternate form of the derivative is given by the following limit: \n\(f'(c) = \lim_{x \to c} (f(x) - f(c)) / (x - c)\)

Substitute the Function and Point into formula

Substitute \(f(x)=3/x\) and \(c=4\) in the alternate form of the derivative: \n\(f'(4) = \lim_{x \to 4} ((3/x - 3/4) / (x - 4))\)

Simplify and Solve the Limit

To simplify this expression, find a common denominator for each fraction and manipulate the expression.\n\(f'(4) = \lim_{x \to 4} ((3x-12) /(4x^2) / (x - 4)) = \lim_{x \to 4} (3x-12)/ (4x(x - 4))\).\nThe fraction simplifies to \(f'(4) = \lim_{x \to 4} ((3/4)(1/x - 3/x^2))\).\nAs \(x\) approaches to \(4,\) the expression inside limit simplifies to \(-3/16.\)

Key concepts

Derivatives

Derivatives play a pivotal role in calculus, often considered the bread and butter of the subject. Fundamentally, a derivative represents the rate at which a function's output changes as its input changes. Think of it as a snapshot capturing the function's behavior at a particular instant.

When you're driving a car, your speedometer gives you a rate of change – your speed – which is the derivative of your position with respect to time. Similarly, in calculus, if you have a function, say, the distance a ball has traveled with time, its derivative gives you the ball's speed at any time point.

In the exercise, finding the derivative of the function \( f(x)=3/x \) at \( x=c \) using the alternate form is akin to determining the function's instantaneous rate of change at that point. A practical use of derivatives can be found in optimizing functions to find the maximum or minimum values, which is immensely useful in fields like economics, engineering, and physics.

Limits

Limits are the foundation of calculus, and they help us deal with values that are not immediately attainable, especially when dealing with continuous variables. Concepts like instant velocity and the slope of a tangent line only make sense through the lens of limits.

For example, when you approach a traffic light that's about to change color, you're dealing with a limit scenario – you assess the situation as you get closer to the light. Similarly, limits in calculus allow you to evaluate what happens to a function as the input approaches a certain value without necessarily reaching that value.

The key to solving the exercise problem lies in understanding the limit used in the alternate form of the derivative, \( \lim_{x \to c} (f(x) - f(c)) / (x - c) \). It captures the essence of change as \( x \) becomes exceedingly close to \( c \), essentially describing the function's tendency around that point.

Calculus

Calculus is an essential mathematical tool that allows us to analyze changes. Divided primarily into the study of derivatives (differential calculus) and integrals (integral calculus), it's a language spoken broadly in the realms of science, economics, and engineering.

Through calculus, you can dissect complex problems into simpler, solvable parts. It could range from calculating areas under curves to understanding the motion of planets. The importance of grasiting fundamental concepts like the limit process, as shown in the alternate form of the derivative, cannot be overstated, as it empowers students to tackle real-world problems that involve constant change.

The exercise provided exemplifies how calculus is used to find precise quantities - derivatives, which have numerous applications such as predicting growth, measuring shapes, and describing physical phenomena in precise mathematical terms.

Rate of Change

The rate of change is a concept that measures how one quantity changes in relation to another. It's present in almost every part of mathematics that deals with movement and growth. In the realm of calculus, when we speak of the 'rate of change', we're mostly referring to derivatives.

When a child slides down a slide, the speed varies at different points - this varying speed is the rate of change of the child's position. In the exercise, finding the derivative at \( x=4 \) for the function \( f(x)=3/x \) with the alternate form reveals the rate at which the function's value is changing at that particular input.

Rates of change can be constant or variable, and understanding them through the use of derivatives provides invaluable insights into the behavior of various phenomena, be it the decay of a radioactive substance or the increase in population of a city.