Problem 33 - Calculus an Applied Approach

Question

Find the value of the derivative of the function at the given point. $$ y=(2 x+1)^{2} \quad(0,1) $$

Step-by-Step Solution

Answer

The value of the derivative of the function \(y=(2x+1)^2\) at the point (0,1) is 4.

Step 1: Differentiate the function using the chain rule.

The derivative of \(y = (2x+1)^2\) can be found using the chain rule. This rule is used to differentiate a function that is composed of another function. In the given equation, \(2x+1\) can be seen as a function within another function \((2x+1)^2\). So in using the chain rule, the power rule is applied first to the outer function which gives \(2*(2x+1)\), afterwards, the derivative of the inner function which is \(2\) is calculated, then both results are multiplied. This gives the derivative as \( y' = 2*(2x+1) * 2 = 4*(2x+1\)

Step 2: Evaluate the derivative at the given point.

Now, plug in the x-coordinate of the given point into the derivative. The point given in the exercise is (0,1), so we will input \(x=0\) into the equation. This gives \( y' = 4*(2*0+1) = 4 \)

Step 3: Interpret the Result

The derivative of the function \(y=(2x+1)^2\) at the point (0,1) is 4. It suggests that at this specific point, for a tiny change in x, y would change 4 times as much. It is an indication of the rate of change of our function at the given point.

Key Concepts

Derivative of a Function

Understanding the derivative of a function is akin to unraveling the DNA of calculus. It fundamentally represents how a function changes as its inputs change. To put it simply, the derivative tells us the rate at which one quantity changes with respect to another.

Take speed, for instance: If you drive a car and press the accelerator, the speedometer shows your speed increasing - that's a rate of change. In calculus, the speedometer is akin to taking the derivative, telling you how fast your position is changing over time. Similarly, in our exercise, finding the derivative of the function \(y=(2x+1)^2\) is crucial to understanding how \(y\) will change when we make tiny changes to \(x\).

Using the derivative, we can analyze the behavior of the function at every point and predict how it will behave for both tiny and large adjustments to its inputs. It's like having a crystal ball for functions, offering us insight into their future behavior based on current trends.

Power Rule

Imagine you're given the task of climbing a stairway where each step is a constant height, like continually raising a function to a power. The power rule in calculus is the tool that lets you jump up these steps with ease.

Applying the power rule is as simple as bringing down the exponent as a multiplier to the base term and then reducing the exponent by one. For example, if our function was \(x^3\), using the power rule, the derivative would be \(3x^2\).

In our textbook problem, the function \(y=(2x+1)^2\) has an exponent of two, which we can gracefully handle with the power rule. The rule tells us to multiply by 2 and then subtract one from our exponent, simplifying the differentiation process and making what might seem like a complex function much more manageable.

Differentiate Composite Functions

When we look at composite functions, we're peering at functions within functions - just like Russian nesting dolls. To differentiate these, we need a special technique that combines our intuition with careful calculation.

This technique is known as the chain rule. Think of it like a production line in a factory: you have to process each part separately before assembling the whole product. The chain rule allows us to take a complex, layered function apart, differentiate each piece, and then put it all back together in an orderly manner.

In our problem, \(y=(2x+1)^2\) is actually a composite function - a function inside another function. The outer function is something squared, and the inside function is \(2x+1\). To differentiate this, we first apply the power rule to the outer function, then multiply it by the derivative of the inner function. It's a dance of sorts, where each function gets its turn to shine and contribute to the final result.

Rate of Change

The concept of 'rate of change' is like feeling the pulse of a function - it gives us the heartbeat, or rhythm, of how a function lives and breathes as its variables shift. In the context of calculus, the rate of change is the measure of how a function's output, typically denoted as \(y\), changes as the input, \(x\), changes a smidge.

By measuring the derivative at a particular point, we capture a snapshot of this rate, which tells us exactly how responsive \(y\) is to minute changes in \(x\). In the case of our initial exercise with \(y = (2x+1)^2\) at the point (0,1), we learned that the rate of change is 4. This means if we were to increase \(x\) by a tiny amount, let's say \(dx\), \(y\) would increase by 4 times that tiny amount, or \(4dx\).

Understanding rate of change has practical implications in everything from physics, where it describes acceleration, to economics, where it tells how quickly costs are rising. It allows us to anticipate and react to the ebb and flow of variables as they interact and evolve over time.