warm-up.md

Warm up: Exponential Functions

1. Calculate with powers

a) $3^n \times 9 = 3^{n+2}$

b) $\frac{\normalsize x\times 4}{\normalsize (2x)^2} = \frac{\normalsize x\times 2^2}{\normalsize x^2 \times 2^2} = \frac{\normalsize x}{\normalsize x^2} = x^{-1}$

c) $\frac{\normalsize 3^4 \times 3}{\normalsize 3^{-2}} = 3^7$

d) $(a^2)^4 \times a^{-3} = a^5$

2. Solve inequations

a) $$2x+4 > 0$$ $$2x > -4$$ $$x > -2$$

b) $$x^2 - 1 < 0$$ $$x^2 < 1$$ $$x < 1$$

c) $$-3x + 4 \le 0$$ $$-3w \le -4$$ $$x \le \frac{4}{3}$$

d) $$x+4 > 3x + 1$$ $$2x > -3$$ $$x > \frac{-3}{2}$$

e) $$x^2 - 4 < - 5$$ $$x^2 > - 1$$ $$x > \emptyset$$

f) $$-7x+1 > -3(x+1)$$ $$-7x+1 > -3x+3$$ $$-4x > -4$$ $$x > 1$$

3. Factorize

a) $3x-x^3 = x(3-x^2)$

b) $9-x^2 = (3-x)^2$

c) $5(x+1) + x^2(-x-1) = 5x+5 - x^3 - x^2 = 5x+5 = 5(x+1)$

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

4. Derivate

5. Tangent

Derivate of $f(x) = x^2 - 2x + 3$ is $f'(x) = 2x-2$

1. $f'(2) = 2$

   $y=2(x-2)+3$

2. $f'(1) = 0$

   $y = 2$

6. Work with geometrical sequences

1. $u_n = 3 \times 2^n$ is geometrical because it's initial term is $3$ and it's common ratio is $3$

2. $(u_n)$ $q=\frac{1}{4}$ $u_0 = 2$

$$u_{n+1} = \frac{n}{4}$$

$$S = \sum_{k=0}^{k=10} u_k$$

$$S = u_0 + u_1 + ... + u_{10}$$

$$S = \frac{2}{1} + \frac{2}{4} + \frac{2}{16} + \frac{2}{64} + ... + \frac{2}{4^{10}}$$

$$S = 2 \times 4^0 + 2 \times 4^{-1} + 2 \times 4^{-2} + ... + 2 \times 4^{-10}$$

$$S = 2 \times (4^0 + 4^{-1} + 4^{-2} + ... + 4^{-10})$$

$$S = 2 \times 1.3333330154 = 2.6666660308$$

7. Evolution and geometrical sequence

a) $q = \frac{1}{3}$

b) $q = 1.10$

c) $q = 0.87$

d) $q = \frac{1}{4}$