To address your request, I will solve the exercises step by step. Let’s begin with Exercise 8B and then move to Exercise 8C.

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Exercise 8B: Differentiation

We are tasked with finding $f'(x)$ for the given functions. I'll solve each one.

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a. $f(x) = 12x + x^2 + 5$

Differentiate term by term: $f'(x) = \frac{d}{dx}(12x) + \frac{d}{dx}(x^2) + \frac{d}{dx}(5)$ $f'(x) = 12 + 2x + 0 = 12 + 2x$

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b. $f(x) = 6x^5 + 6x - 7$

Differentiate term by term: $f'(x) = \frac{d}{dx}(6x^5) + \frac{d}{dx}(6x) + \frac{d}{dx}(-7)$ $f'(x) = 30x^4 + 6 + 0 = 30x^4 + 6$

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c. $f(x) = \frac{1}{2}x^{-1} - \frac{1}{2}x^{-2}$

Differentiate term by term: $f'(x) = \frac{d}{dx}\left(\frac{1}{2}x^{-1}\right) + \frac{d}{dx}\left(-\frac{1}{2}x^{-2}\right)$ $f'(x) = -\frac{1}{2}x^{-2} + x^{-3} = -\frac{1}{2x^2} + \frac{1}{x^3}$

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d. $f(x) = 4x^3 + 3x^{1/2} + 10x^{-1}$

Differentiate term by term: $f'(x) = \frac{d}{dx}(4x^3) + \frac{d}{dx}(3x^{1/2}) + \frac{d}{dx}(10x^{-1})$ $f'(x) = 12x^2 + \frac{3}{2}x^{-1/2} - 10x^{-2}$

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e. $f(x) = 4 - 12x^{-2} + 4x^{1/2}$

Differentiate term by term: $f'(x) = \frac{d}{dx}(4) + \frac{d}{dx}(-12x^{-2}) + \frac{d}{dx}(4x^{1/2})$ $f'(x) = 0 + 24x^{-3} + 2x^{-1/2}$

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f. $f(x) = 5x^{3/2} + 10x^{-2}$

Differentiate term by term: $f'(x) = \frac{d}{dx}(5x^{3/2}) + \frac{d}{dx}(10x^{-2})$ $f'(x) = \frac{15}{2}x^{1/2} - 20x^{-3}$

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g. $f(x) = 9x^2 + x^2 + 4x + x^{-1}$

Combine like terms first: $f(x) = 10x^2 + 4x + x^{-1}$

Differentiate term by term: $f'(x) = \frac{d}{dx}(10x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(x^{-1})$ $f'(x) = 20x + 4 - x^{-2}$

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h. $f(x) = 5x^{-2} + x^{-3} - 2x^{-1}$

Differentiate term by term: $f'(x) = \frac{d}{dx}(5x^{-2}) + \frac{d}{dx}(x^{-3}) + \frac{d}{dx}(-2x^{-1})$ $f'(x) = -10x^{-3} - 3x^{-4} + 2x^{-2}$

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Exercise 8C: Integration

Now, we will solve the integrals provided.

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1. $\int (x^3 + 2x) \, dx$

Integrate term by term: $\int x^3 \, dx + \int 2x \, dx$ $\frac{x^4}{4} + x^2 + C$

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2. $\int (2x^{-2} + 3) \, dx$

Integrate term by term: $\int 2x^{-2} \, dx + \int 3 \, dx$ $-2x^{-1} + 3x + C$

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3. $\int (5x^{3/2} - 3x^2) \, dx$

Integrate term by term: $\int 5x^{3/2} \, dx - \int 3x^2 \, dx$ $\frac{10}{5}x^{5/2} - x^3 + C$

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4. $\int (2x^{1/2} - 2x^{-1/2} + 4) \, dx$

Integrate term by term: $\int 2x^{1/2} \, dx - \int 2x^{-1/2} \, dx + \int 4 \, dx$ $\frac{4}{3}x^{3/2} - 4x^{1/2} + 4x + C$

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5. $\int (4x^3 - 3x^{-4} + r) \, dx$

Integrate term by term (treat $r$ as a constant): $\int 4x^3 \, dx - \int 3x^{-4} \, dx + \int r \, dx$ [ x^4 + \frac{3}{-3x^3} + rx + C = x^4 + r(x)Would you like me to continue solving the rest of Exercise 8C or clarify any of the above steps? Let me know! Below are some questions to deepen your understanding:

1. What is the difference between integration and differentiation?
2. How do you apply the power rule for derivatives and integrals?
3. Why do we add a constant $C$ during indefinite integration?
4. Can you simplify fractional exponents, such as $x^{3/2}$, before or after differentiation/integration?
5. What are some practical applications of integration in real-world scenarios?

Tip: Always rewrite complex terms with fractional or negative exponents before performing calculus operations. This simplifies the process and reduces errors.