Chapter 6: Problem 5

Calculate \(f(x+h)\) when (a) \(f(x)=x^{2}\) (b) \(f(x)=x^{3}\) (c) \(f(x)=\frac{1}{x}\) In each case write down the corresponding expression for \(f(x+h)-f(x)\).

Short Answer

Expert verified

Question: Calculate \(f(x+h)\) and the expression for \(f(x+h)-f(x)\) for the following functions: a) \(f(x) = x^2\) b) \(f(x) = x^3\) c) \(f(x) = \frac{1}{x}\) Answer: a) For \(f(x) = x^2\), \(f(x+h) = x^2 + 2xh + h^2\) and \(f(x+h) - f(x) = 2xh + h^2\). b) For \(f(x) = x^3\), \(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\) and \(f(x+h) - f(x) = 3x^2h + 3xh^2 + h^3\). c) For \(f(x) = \frac{1}{x}\), \(f(x+h) = \frac{1}{x+h}\) and \(f(x+h) - f(x) = \frac{-h}{x(x+h)}\).

Step by step solution

(Step 1: Calculate f(x+h) for f(x) = x^2)

Substitute \(x\) with \(x+h\) in the function \(f(x)=x^2\). Then, \(f(x+h)=(x+h)^2\) Now expand the expression: \(f(x+h) = x^2 + 2xh + h^2\)

(Step 2: Calculate f(x+h) - f(x) for f(x) = x^2)

Subtract \(f(x)\) from \(f(x+h)\): \(f(x+h)-f(x) = (x^2 + 2xh + h^2) - x^2\) Simplify the expression: \(f(x+h)-f(x) = 2xh + h^2\) For the function \(f(x)=x^2\), \(f(x+h)=x^2+2xh+h^2\) and \(f(x+h)-f(x) = 2xh + h^2\).

(Step 3: Calculate f(x+h) for f(x) = x^3)

Substitute \(x\) with \(x+h\) in the function \(f(x)=x^3\). Then, \(f(x+h)=(x+h)^3\) Expand the expression: \(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\)

(Step 4: Calculate f(x+h) - f(x) for f(x) = x^3)

Subtract \(f(x)\) from \(f(x+h)\): \(f(x+h)-f(x) = (x^3 + 3x^2h + 3xh^2 + h^3) - x^3\) Simplify the expression: \(f(x+h)-f(x) = 3x^2h + 3xh^2 + h^3\) For the function \(f(x)=x^3\), \(f(x+h)=x^3+3x^2h+3xh^2+h^3\) and \(f(x+h)-f(x) = 3x^2h + 3xh^2 + h^3\).

(Step 5: Calculate f(x+h) for f(x) = 1/x)

Substitute \(x\) with \(x+h\) in the function \(f(x)=\frac{1}{x}\). Then, \(f(x+h)=\frac{1}{x+h}\)

(Step 6: Calculate f(x+h) - f(x) for f(x) = 1/x)

Subtract \(f(x)\) from \(f(x+h)\): \(f(x+h)-f(x) = \frac{1}{x+h} - \frac{1}{x}\) Find a common denominator to combine the fractions: \(f(x+h)-f(x) = \frac{x-(x+h)}{x(x+h)}\) Simplify the expression: \(f(x+h)-f(x) = \frac{-h}{x(x+h)}\) For the function \(f(x)=\frac{1}{x}\), \(f(x+h)=\frac{1}{x+h}\) and \(f(x+h)-f(x) = \frac{-h}{x(x+h)}\).

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Calculate \(f(x+h)\) when (a) \(f(x)=x^{2}\) (b) \(f(x)=x^{3}\) (c) \(f(x)=\frac{1}{x}\) In each case write down the corresponding expression for \(f(x+h)-f(x)\).

Short Answer

Step by step solution

(Step 1: Calculate f(x+h) for f(x) = x^2)

(Step 2: Calculate f(x+h) - f(x) for f(x) = x^2)

(Step 3: Calculate f(x+h) for f(x) = x^3)

(Step 4: Calculate f(x+h) - f(x) for f(x) = x^3)

(Step 5: Calculate f(x+h) for f(x) = 1/x)

(Step 6: Calculate f(x+h) - f(x) for f(x) = 1/x)

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