Tutorial Week 9
This tutorial covers L’Hopital’s rule, linearization, and Mean Value Theorem.
L’Hopital’s Rule
Q1: Compute \(\lim_{x \to 0} \frac{x + sin(x)}{x^2 + cos(x)tan(x)}\).
Q2: Compute \(\lim_{x \to 0^+} xln(x)\).
Q3: Compute \(\lim_{x \to 0^{+}} \frac{ln(x)}{log_3(2x)}\).
Q4: Compute \(\lim_{x \to 1} (\frac{x}{x-1} - \frac{1}{ln(x)})\).
L’Hopital’s Rule with Logs
Q5: Compute \(\lim_{x \to \infty} \left(\frac{1}{e^{x}}+1\right)^{e^{x}}\).
Mean Value Theorem
Q6: Given \(f(1) = 10\), \(f'(x) \geq 3\) for \(x \in [1, 5]\), what would be the highest lower bound for \(f(5)\)?
Q7: If \(3 \leq f'(x) \leq 6\) for all \(x\), show that \(18 \leq f(7) − f(1) \leq 36\).
Q8: Show that \(4x+2 = \cos\left(x\right)\) has exactly one solution.
Linearization
Q9: Find the linearization of \(f(x)=3\sqrt{x} + 4\) at \(x=16\).
Q10: Find the linear approximation of \(f(x) = tan(x)\) at \(x=\pi\).
Q11: Find the linear approximation of \(f(x) = 3e^{x}\) at \(x=0\). Use it to approximate \(3e^{0.2}\).
