Question

Multiple Choice Let \(f(x)=4-3 x\) . Which of the following is equal to \(f^{\prime}(-1) ? \)(\mathbf{A})-7 \quad(\mathbf{B}) 7 \quad(\mathbf{C})-3 \quad(\mathbf{D}) 3 \quad(\mathbf{E})$ does not exist

Short answer

The derivative of the function at \(x=-1\) is -3, hence, Option C is the correct choice.

Step-by-step solution

Identify the derivative of \(f(x)\)

Derivative of a constant is zero and derivative of \(x\) is one. So if we apply these rules to \(f(x)=4-3x\) we get \(f'(x) = 0 - 3 = -3\)

Evaluate \(f'(x)\) at \(x = -1\)

Since the derivative is constant, its value is -3 for all points in its domain, including \(x = -1\) Hence, \(f'(-1) = -3\)

Key concepts

Calculus

Calculus is an area of mathematics that studies how things change. It gives us the tools to calculate the rate of change—and this is where derivatives come into play. Derivatives represent the rate at which a function is changing at any given point. Think of it as an exact way to describe how an object's speed might change from moment to moment. When working with functions in calculus, we're often interested in finding this precise rate of change, rather than just making an estimate.

In the context of the exercise, calculus helps us understand the problem by providing a systematic approach to find the rate of change of the linear function given. The derivative is the tool we used to determine this rate at a specific point, and even more generally, it helps us describe the behavior of the function across its entire domain.

Derivatives

Derivatives are the fundamental building blocks in calculus that measure how a function's output changes as its input changes. Think of derivatives as the mathematical equivalent of a speedometer that adjusts as a car speeds up or slows down; they tell us the 'speed' of a function at any given moment.

To find the derivative of a function, we look for the limit of the average rate of change as the change in the input approaches zero. For linear functions, like the one given in the exercise, the derivative is simply the coefficient of the input variable. In our case, for the function \(f(x) = 4 - 3x\), the derivative \(f'(x) = -3\) indicates that for every unit increase in \(x\), the function \(f\) decreases by 3 units.

Function Notation

Function notation is a way of representing functions algebraically, which is both concise and convenient for performing a variety of mathematical operations. In the notation \(f(x)\), \(f\) represents the function and \(x\) represents the input to the function. This allows us to work with functions abstractly without always needing to refer to specific inputs or outputs.

In the exercise, the function notation \(f'(x)\) is used to represent the derivative of function \(f\). The notation \(f'(-1)\) tells us specifically to evaluate the derivative of \(f\) when \(x\) is -1, which—in this case—gives us a constant output of -3. Understanding this notation is key to interpreting calculus problems and conveying the solutions clearly.

Slope of a Line

In algebra and calculus, the slope of a line is a numerical measure of its steepness, often denoted as 'rise over run'. It is the change in the y-coordinate divided by the corresponding change in the x-coordinate between two distinct points on the line. With straight lines, this value is constant; however, with curves, the slope changes from point to point.

When we talk about the derivative of a linear function, we're actually discussing the slope of the line it represents. For the function \(f(x) = 4 - 3x\), the derivative \(f'(x)\) is -3, meaning the slope of the line is -3. This is consistent through any two points on the line, signifying a decrease in the y-value by 3 units for every increase in x by 1 unit. The ability to find the slope and understand its implications is essential for analyzing functions and modeling real-world phenomena.