In Exercises 33–34, find and simplify the difference quotient [f(x + h) - f(x)]/h, h =/= 0 for the given function. f(x) = -2x^2 + x + 10

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Step 1: Start by substituting (x + h) into the function f(x). The function is f(x) = -2x^2 + x + 10, so f(x + h) becomes f(x + h) = -2(x + h)^2 + (x + h) + 10.

Step 2: Expand the expression for f(x + h). Use the distributive property to expand (x + h)^2, which gives x^2 + 2xh + h^2. Substituting this back, f(x + h) = -2(x^2 + 2xh + h^2) + x + h + 10.

Step 3: Simplify f(x + h) by distributing -2 and combining like terms. This results in f(x + h) = -2x^2 - 4xh - 2h^2 + x + h + 10.

Step 4: Compute the difference f(x + h) - f(x). Subtract f(x) = -2x^2 + x + 10 from the expanded f(x + h). This gives [-2x^2 - 4xh - 2h^2 + x + h + 10] - [-2x^2 + x + 10].

Step 5: Simplify the numerator of the difference quotient by canceling out terms and combining like terms. Then divide the simplified numerator by h (noting that h ≠ 0). The final expression will be the simplified form of the difference quotient.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as [f(x + h) - f(x)]/h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the derivative, which measures the instantaneous rate of change.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of the difference quotient, you need to evaluate the function f at both x and x + h. This step is crucial for calculating the difference quotient accurately and understanding how the function behaves as x changes.

Simplification of Expressions

Simplification of expressions is the process of reducing a mathematical expression to its simplest form. This often involves combining like terms, factoring, or canceling common factors. In the context of the difference quotient, simplifying the resulting expression after substituting into the formula is essential for finding a clear and concise representation of the average rate of change.

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