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Question One: Sequence Convergence and Bounds

1. (a) Show that the sequence $u_n = \frac{1}{4} + \frac{1}{4^{n+1}}$ is bounded and monotonic.

   • Approach: Prove that $u_n$ is bounded by finding its upper and lower bounds. Then, show it’s monotonic by verifying if $u_{n+1} \leq u_n$ (monotonically decreasing) or $u_{n+1} \geq u_n$ (monotonically increasing).

2. (b) Show that the sequence $x_n = (2x_n + 5)/(x_n + 4)$ converges.

   • Approach: Check if $x_n$ has a fixed point by solving $x = \frac{2x + 5}{x + 4}$ for $x$. Then, determine if $x_n$ approaches this value as $n \to \infty$.

Question Two: Limits and Function Ranges

1. (a) Show that $x^2 + 2 \sin x = x$ has at least one solution in the interval $[0, 1]$.

   • Approach: Use the Intermediate Value Theorem. Evaluate the function at the endpoints of the interval to see if there is a sign change, which indicates a root.

2. (b) Describe the range of the function $f(x) = 2 + (q - x) \sin (x - \pi)$.

   • Approach: Analyze the behavior of $f(x)$ based on the value of $q$ and the periodicity of $\sin(x - \pi)$.

Question Three: Differentiation and Function Analysis

1. (a) Find the coordinate point $x$ for which $y = \frac{3x}{x^2 + 1} - x^2 + 4$ has a horizontal tangent.

   • Approach: Set the derivative $\frac{dy}{dx} = 0$ and solve for $x$.

2. (b) Find the differential coefficient of $f(x) = 3x \sin 2x$.

   • Approach: Differentiate using the product rule, $f'(x) = (3x)' \sin 2x + 3x (\sin 2x)'$.

Question Four: Advanced Differentiation and Integrals

1. (a) Differentiate $y = (x + 2)(x + 1)(x + 3)(x + 2)$.

   • Approach: Expand the expression or use the product rule multiple times to differentiate.

2. (b) Find $f(x) / f'(x)$ if $f(x) = \frac{x^2 + 1}{x^3 + x^2}$.

   • Approach: First find $f'(x)$ by applying the quotient rule, then divide $f(x)$ by $f'(x)$.

3. (c) Find the derivative of $y$ if $y = \frac{x^{1.7}}{2} - \sin x + \frac{5}{x}$.

   • Approach: Differentiate each term separately: use power rule, derivative of sine, and the power rule for $x^{-1}$.

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Further Questions to Consider

1. How does the Intermediate Value Theorem guarantee at least one solution in a given interval?
2. What is the importance of finding fixed points in determining the convergence of sequences?
3. How do horizontal tangents relate to critical points in calculus?
4. What role does monotonicity play in ensuring the convergence of sequences?
5. Can you apply the quotient rule to other functions similar to $f(x) = \frac{x^2 + 1}{x^3 + x^2}$?

Tip:

When differentiating products or quotients, remember to apply the product and quotient rules carefully, ensuring that each step is clearly defined to avoid errors.