Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
e⁰·⁰⁶

Verified step by step guidance

1. Identify the function you want to approximate. In this case, the function is f(x) = e^x.
2. Choose a value of 'a' that is close to the point you want to estimate, which is x = 0.06. A good choice for 'a' is 0, because e^0 = 1 and it simplifies calculations.
3. Find the derivative of the function f(x) = e^x. The derivative, f'(x), is also e^x.
4. Use the formula for linear approximation: L(x) = f(a) + f'(a)(x - a). Substitute a = 0, f(a) = e^0 = 1, and f'(a) = e^0 = 1 into the formula.
5. Calculate L(0.06) using the linear approximation formula: L(0.06) = 1 + 1 * (0.06 - 0). This will give you an estimate for e^0.06.

Key Concepts

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that a function can be closely approximated by a linear function when the input is near a specific value. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where 'a' is the point of tangency.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function at that point. For linear approximations, the derivative is crucial as it determines the slope used in the linear equation.

Exponential Function

The exponential function, denoted as e^x, is a mathematical function that grows rapidly as x increases. It is defined as the function whose derivative is equal to itself, making it unique in calculus. Understanding the properties of the exponential function is essential for estimating values like e^0.06, especially when using linear approximations around a point like e^0.