Math Problem Statement

Find the derivative of the following functions using the limit definition.

Solution

The assignment asks to find the derivative of each function using the limit definition. The limit definition of the derivative is given by:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Let's go through each function step-by-step and apply the limit definition.

1. f(x)=7f(x) = 7

For a constant function, the derivative is 0, since there is no change in the function value as xx changes.

f′(x)=lim⁡h→0f(x+h)−f(x)h=lim⁡h→07−7h=0f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{7 - 7}{h} = 0

So, f′(x)=0f'(x) = 0.

2. f(x)=7x−3f(x) = 7x - 3

This is a linear function. Using the limit definition:

f′(x)=lim⁡h→0(7(x+h)−3)−(7x−3)hf'(x) = \lim_{h \to 0} \frac{(7(x+h) - 3) - (7x - 3)}{h}

= \lim_{h \to 0} \frac{7h}{h} = 7$$ So, $$f'(x) = 7$$. ### 3. $$f(x) = 3 + \frac{2}{3}x$$ This is also a linear function. Applying the limit definition: $$f'(x) = \lim_{h \to 0} \frac{\left(3 + \frac{2}{3}(x+h)\right) - \left(3 + \frac{2}{3}x\right)}{h}$$ $$= \lim_{h \to 0} \frac{3 + \frac{2}{3}(x+h) - 3 - \frac{2}{3}x}{h} = \lim_{h \to 0} \frac{\frac{2}{3}h}{h} = \frac{2}{3}$$ Thus, $$f'(x) = \frac{2}{3}$$. ### 4. $$f(x) = x^2 + x - 3$$ This is a quadratic function. Applying the limit definition: $$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 + (x+h) - 3 - (x^2 + x - 3)}{h}$$ $$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 + x + h - 3 - x^2 - x + 3}{h}$$ $$= \lim_{h \to 0} \frac{2xh + h^2 + h}{h} = \lim_{h \to 0} (2x + h + 1) = 2x + 1$$ Thus, $$f'(x) = 2x + 1$$. ### 5. $$f(x) = x^3 - 12x$$ This is a cubic function. Using the limit definition: $$f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - 12(x+h) - (x^3 - 12x)}{h}$$ $$= \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - 12x - 12h - x^3 + 12x}{h}$$ $$= \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - 12h}{h} = \lim_{h \to 0} (3x^2 + 3xh + h^2 - 12) = 3x^2 - 12$$ Thus, $$f'(x) = 3x^2 - 12$$. --- This covers the first five problems. Would you like me to continue with the remaining ones? Or would you prefer additional details on these calculations? Here are five related questions you could explore: 1. What is the geometric interpretation of the derivative of a constant function? 2. How can you derive the formula for the derivative of a power function $$x^n$$? 3. Why do linear functions have constant derivatives? 4. How does the derivative change for polynomial functions of higher degrees? 5. How can we compute the derivative of a function involving fractional powers? **Tip:** When applying the limit definition of the derivative, carefully expand all terms and combine like terms to simplify the expression for easier limit evaluation.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Limits

Formulas

f'(x) = lim(h -> 0) [f(x+h) - f(x)] / h

Theorems

Limit Definition of the Derivative

Suitable Grade Level

Undergraduate Calculus or High School AP Calculus

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