1. The problem is to understand and apply key concepts in functions, trigonometry, differential calculus, and hypothesis testing, including definitions, rules, and examples. 2. Let's start with **Function Definition and Domain**. - A function $f$ from set $A$ to set $B$ assigns each element in $A$ exactly one element in $B$. - The domain is the set of all possible inputs. **Example 1:** Find the domain of $f(x) = \frac{1}{x-3}$. - The denominator cannot be zero, so $x-3 \neq 0 \Rightarrow x \neq 3$. - Domain: $\{x \in \mathbb{R} : x \neq 3\}$. 3. **Trigonometry - Angle Conversion** - To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$. - To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$. **Example 2:** Convert $60^\circ$ to radians. - $60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. 4. **Differential Calculus: Definition of a Derivative** - The derivative of $f$ at $x$ is $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$. **Example 3:** Find the derivative of $f(x) = x^2$. - $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x$. 5. **Differentiation Rules** - Power rule: $\frac{d}{dx} x^n = nx^{n-1}$. - Sum rule: derivative of sum is sum of derivatives. 6. **Derivatives of Trigonometric Functions** - $\frac{d}{dx} \sin x = \cos x$. - $\frac{d}{dx} \cos x = -\sin x$. 7. **Applications of Derivatives** - Slope of tangent line at $x=a$ is $f'(a)$. - Equation of tangent line: $y = f(a) + f'(a)(x - a)$. **Example 4:** Find the tangent line to $f(x) = x^2$ at $x=1$. - $f(1) = 1$, $f'(x) = 2x$, so $f'(1) = 2$. - Equation: $y = 1 + 2(x - 1) = 2x - 1$. 8. **Hypothesis Testing: z-test and t-test** - z-test is used when population variance is known and sample size is large. - t-test is used when population variance is unknown and sample size is small. **Example 5:** z-test for mean with $\alpha=0.05$, sample mean $\bar{x}=105$, population mean $\mu=100$, population std dev $\sigma=10$, sample size $n=36$. - Test statistic: $z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{105 - 100}{10/6} = 3$. - Since $z=3 > 1.96$, reject null hypothesis. This covers the first problem with examples and explanations.