If f is a continuous function with f(2)=5, and if the slope of the tangent line to the graph of

step 1 So in this problem, we are given that at x equals 2, the tangent line is y equals 5x minus 3, an

If f is a continuous function with f(2)=5, and if the slope...

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Key Concepts

Derivative as Slope

The derivative of a function at a point represents the slope of its tangent line. In this problem, the provided slopes in different intervals tell us the value of the derivative of f over those intervals, which is a central concept in differentiating and understanding rate of change in calculus.

Antiderivatives and Integration

Knowing the derivative of a function allows us to recover the function itself by finding its antiderivative or integral. In this problem, integrating the constant slopes over their respective intervals, with an additional constant of integration, is the method used to reconstruct the function f.

Continuity in Piecewise Functions

Since the function f is continuous, the pieces obtained from integrating the derivative in different intervals must meet without any jumps or breaks. This fact is essential for determining the correct integration constants in the separate intervals to form a single continuous function.

Initial Value Problem

An initial value problem involves determining a specific solution to a differential equation that satisfies a given condition, here f(2) = 5. This condition is used to calculate the constants of integration when finding the antiderivative, ensuring that the resulting function f fits the given criteria.

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