Problem 23 - Single Variable Calculus

Question

Find the rate of change of \(y\) with respect to \(x\) at the given value of \(x\). \(y=-2 x^{2}+x+1 ; \quad x=1\)

Step-by-Step Solution

Answer

The rate of change of \(y\) with respect to \(x\) at \(x = 1\) is -3.

Step 1: Find the derivative of the function with respect to x

To find the derivative of the function, we will apply the power rule, which states that if \(y = ax^n\), where a and n are constants, then \(y' = anx^{n-1}\). Applying this rule to our given function, we get: \(y' = \frac{d}{dx}(-2x^2 + x + 1)\) For each term in the function, apply the power rule: \(y' = \frac{d}{dx}(-2x^2) + \frac{d}{dx}(x) + \frac{d}{dx}(1)\) Calculating the derivative of each term: \(y' = -4x + 1 + 0\) So, the derivative of the given function is: \(y' = -4x + 1\)

Step 2: Evaluate the derivative at x = 1

Now that we have the derivative, we need to find the rate of change (value of the derivative) at \(x = 1\). To do this, substitute \(x = 1\) into the derivative and solve: \(y'(1) = -4(1) + 1\) Calculating the value: \(y'(1) = -4 + 1\) \(y'(1) = -3\) The rate of change of y with respect to x at \(x = 1\) is -3.

Key Concepts

Derivative

The derivative is a fundamental tool in calculus that measures how a function changes as its input changes. It's like the mathematical equivalent of finding the speed of a car at a specific moment during a trip. Essentially, a derivative represents an instantaneous rate of change.

In the context of our exercise, the derivative tells us how quickly the value of the function 'y' is changing at a particular value of 'x'. By finding the derivative of a function, you can make predictions about the function's behavior, such as increasing and decreasing intervals, which is crucial in many fields like physics and economics.

When working with derivatives, it's crucial to understand different notations like the prime notation (\(f'(x)\)) or the Leibniz notation (\frac{dy}{dx}). Both notations express the derivative of a function but may be used in different contexts or for emphasizing different aspects of differentiation.

Power Rule

Differentiating polynomial functions can be made simple with the power rule. This rule swiftly calculates the derivative of any term with the form \(ax^n\), where 'a' is a coefficient and 'n' is a positive integer exponent.

According to the power rule, to find the derivative of \(ax^n\), simply multiply the exponent by the coefficient (giving \(an\)) and subtract one from the exponent. The result is \(anx^{n-1}\).

If we apply this to \(x^2\), for instance, the derivative would be \(2x^{2-1} = 2x\). What's handy about the power rule is that it works for any term that is a power of 'x', even when 'x' is raised to the first power or zero, which corresponds to the function \(x\) and a constant term respectively.

Differentiation

Differentiation is the process of finding a derivative. It provides a systematic way to compute the rate at which a quantity changes. When you differentiate a function, you're finding a new function that gives the slope of the tangent line to the original function at any point.

There are several rules and techniques for differentiation, including the product rule, quotient rule, chain rule, and more. However, the power rule, as we've discussed, is one of the most straightforward methods for dealing with polynomials.

Differentiation is more than a mechanical process; it's a powerful tool that helps us model and understand real-world phenomena where change is a constant factor. For example, in economics, it's used to find marginal costs and revenues, and in physics, to describe objects in motion.

Single Variable Calculus

Single variable calculus is the branch of calculus focusing on functions of one variable. It encompasses the study of limits, derivatives, integrals, and series. In this context, the variable 'x' in our exercise acts alone and is the independent variable that drives changes in 'y'.

The beauty of single variable calculus lies in its ability to analyze and model problems in one dimension. Whether you're calculating the slope of a curve, the area under a graph, or the accumulated value of a changing quantity over time, single variable calculus provides the necessary mathematical apparatus. It's the starting point for many students in their journey through the fascinating world of calculus.