91Ó°ÊÓ

In calculus, the derivative function represents a fundamental concept, capturing the idea of how a function changes at any given point. In essence, it's the rate of change or the function's instantaneous velocity. For a visual understanding, imagine you're driving and your speedometer reads 60 mph – that's similar to a derivative; it shows how fast you're going at that very moment.

The derivative function, often denoted as f'(x), tells us the slope of the tangent line to the curve of the original function f(x) at any point x. When solving calculus problems involving the derivative, like in our exercise, we first identify the derivative function from the conditions given, such as the slope of the tangent line. Here, we determined that f'(x) = 2e^x + 1.

Application

Understanding the concept of a derivative allows you to determine things like acceleration if you're dealing with a position function in physics, or marginal cost when you're considering a cost function in economics. In the problem at hand, knowing that the slope of the tangent line equals the derivative informs us how to construct the original function using integration.

The slope of a tangent line to a curve at a particular point gives us great insight into the behavior of the function at that point. It effectively measures the steepness of the curve and can tell us whether the function is increasing or decreasing, and at what rate, at a given point. It's like assessing a hill's incline while hiking; a steeper slope means a more significant change in height over a short distance.

When given the slope of the tangent line, like in our exercise with 2e^x + 1, we're given a direct formula for the derivative of the function. This slope function f'(x) serves as a crucial link between the original function and its rate of change at any point along the curve.

Connecting to Derivatives

Since the derivative of a function at a point is exactly the slope of the tangent line at that point, we use this connection to move forward in solving calculus problems. If we understand the slope concept, we can reverse engineer to find the original function by integrating the derivative, as seen in our example problem.

Integration can be thought of as the reverse process of taking a derivative, often referred to as 'anti-differentiation.' While a derivative represents a rate of change, integration involves the accumulation of quantities, such as areas under a curve. Imagine filling a pool with water; integration would give you the total amount of water added over time, while the derivative would tell you the rate at which water flows into the pool at any given moment.

In our example, after acquiring the derivative function f'(x), we performed integration to find the original function f(x). By integrating each term of f'(x) = 2e^x + 1 individually, we get f(x) = 2e^x + x + C. Always remember to add the constant of integration, C, because anti-differentiation is not unique without it.

Finding the Constant

Lastly, to pin down the specific function asked for in the problem, we use initial conditions given, like in our case f(1) = -1, to solve for C. This final step ensures our integrated function perfectly aligns with the specific situation described in the problem.