1. **Problem:** Evaluate the limit $$\lim_{\Delta x \to 0} \frac{\Delta x}{\Delta x}$$. Since for all $$\Delta x \neq 0$$, $$\frac{\Delta x}{\Delta x} = 1$$, the limit as $$\Delta x \to 0$$ is 1. 2. **Problem:** Evaluate the limit $$\lim_{x \to 0} (1 + x)^{\frac{1}{x}}$$. This is a classic limit known to equal $$e$$ because $$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$$. 3. **Problem:** Find the derivative $$\frac{d}{dx} \cot x$$. Using derivative rules, $$\frac{d}{dx} \cot x = -\csc^2 x$$. 4. **Problem:** What is the second derivative commonly called? The second derivative of position with respect to time is called acceleration. 5. **Problem:** What physical quantity is acceleration proportional to? By Newton's second law, acceleration is proportional to the total force acting on an object. 6. **Problem:** Determine if the equation $$x^2 y + y^3 = 4$$ is implicit or explicit. Since $$y$$ cannot be written purely as a function of $$x$$ without rearrangement, it is an implicit equation. 7. **Problem:** Is Taylor series exact or approximate? The Taylor series is generally an approximation of a function near a point. 8. **Problem:** What is a line that touches the graph of a function at two points called? A line that intersects a curve at two points is called a secant line. **Final Answers:** 1) a) 1 2) d) e 3) c) -cosec^2 x 4) b) acceleration 5) a) force 6) a) implicit 7) b) approximate 8) b) secant line