Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.4.18

Capitolo 3, Problema 3.4.18

Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).

Guida verificata passo dopo passo

Step 1: To find the slope of the graph of a function at a given point, we need to find the derivative of the function, f(x). The derivative, f'(x), represents the slope of the tangent line to the graph at any point x.

Step 2: The function given is f(x) = 2 + xe^x. We need to differentiate this function with respect to x. Use the product rule for differentiation, which states that if you have a function u(x)v(x), its derivative is u'(x)v(x) + u(x)v'(x).

Step 3: Identify u(x) = x and v(x) = e^x. Differentiate both: u'(x) = 1 and v'(x) = e^x. Apply the product rule: the derivative of xe^x is 1 * e^x + x * e^x.

Step 4: Combine the derivatives: f'(x) = 0 + (1 * e^x + x * e^x) = e^x + xe^x. The derivative of the constant 2 is 0, so it does not affect the derivative.

Step 5: Evaluate the derivative at the point x = 0 to find the slope at (0, 2). Substitute x = 0 into f'(x) = e^x + xe^x to find the slope at that point.

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Derivative

The derivative of a function at a given point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, finding the slope of the graph at a specific point involves calculating the derivative of the function and evaluating it at that point.

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Derivatives

Exponential Function

An exponential function is a mathematical function of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828). In the given function f(x) = 2 + xe^x, the term xe^x combines polynomial and exponential behavior, which affects the function's growth rate and curvature. Understanding how to differentiate such functions is crucial for finding slopes.

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Exponential Functions

Point-Slope Form

The point-slope form of a linear equation is used to describe the slope of a line given a point on the line. It is expressed as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. In this problem, after finding the derivative, we can use the slope at the point (0, 2) to understand the behavior of the function near that point.

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Slope-Intercept Form

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