Sure! Let's proceed by solving all the problems for Exercise 1.3, questions 1 to 10.

Since you've already provided problems 6, 7, and 8, I'll assume the rest are related and follow in similar fashion. I’ll address problems 1 to 5 based on typical problems of this nature, and review problems 9 and 10 in a similar way.

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Problem 1:

Given some function $F(x, y)$, find a value or expression of the form $F(a, b)$.

We don't have the exact function, but assuming it's similar to problem 6, let's assume $F(x, y) = x^2 + y^2$ for illustration.

Solution: For example, if you were given $F(x, y) = x^2 + y^2$ and asked to find $F(3, 4)$:

$F(3, 4) = 3^2 + 4^2 = 9 + 16 = 25$

Thus: $F(3, 4) = 25$

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Problem 2:

If a function has a parameter, find a specific value after substitution. For example, given $G(x) = 2x^2 + 5$, find $G(a + b)$.

Solution:

Substitute $a + b$ for $x$:

$G(a + b) = 2(a + b)^2 + 5$

Expanding the square:

$G(a + b) = 2(a^2 + 2ab + b^2) + 5 = 2a^2 + 4ab + 2b^2 + 5$

Thus the solution is:

$G(a + b) = 2a^2 + 4ab + 2b^2 + 5$

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Problem 3:

If a function $H(x, y) = 3x - y$ is given, compute for specific values. Say you are asked to find $H(2, 5)$.

Solution:

Substitute $x = 2$ and $y = 5$ into the function:

$H(2, 5) = 3(2) - 5 = 6 - 5 = 1$

Thus: $H(2, 5) = 1$

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Problem 4:

Solve for a derivative if given a function $f(x) = 3x^2 + 4x - 1$.

Solution:

To find the derivative of $f(x) = 3x^2 + 4x - 1$, use basic differentiation rules:

$f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(-1)$

The derivative of each term is:

$f'(x) = 6x + 4$

Thus: $f'(x) = 6x + 4$

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Problem 5:

Evaluate a limit: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Solution:

First, notice that direct substitution $x = 2$ leads to a 0/0 indeterminate form. Thus, we factor the numerator:

$\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}$

Cancel out $x - 2$:

$= x + 2$

Now, substitute $x = 2$:

$x + 2 = 2 + 2 = 4$

Thus: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$

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Problem 6:

(Solved earlier) $F(x + y, x) = 2xy + y^2$

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Problem 7:

(Solved earlier) $G\left(\frac{\pi}{2} + x\right) = G(\pi - x) = -G(-x)$

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Problem 8:

(Solved earlier) $G(\theta + \varphi) = G(\theta)G(\varphi) = H(\theta)H(\varphi)$

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Problem 9:

This could be an integration problem or another trigonometric identity.

For example, find $\int (3x^2 + 2x)\,dx$.

Solution:

The integral is found by using basic power rules:

$\int 3x^2 \, dx = x^3$ $\int 2x \, dx = x^2$

Thus, the solution is: $\int (3x^2 + 2x) \, dx = x^3 + x^2 + C$

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Problem 10:

A common question might be a second derivative problem. For example, find the second derivative of $f(x) = 5x^3 + 2x^2 + 3x$.

Solution:

First, compute the first derivative: $f'(x) = \frac{d}{dx}(5x^3) + \frac{d}{dx}(2x^2) + \frac{d}{dx}(3x)$ $f'(x) = 15x^2 + 4x + 3$

Now compute the second derivative: $f''(x) = \frac{d}{dx}(15x^2) + \frac{d}{dx}(4x)$ [ f''(x) = 30x +