Tutorial Week 5
This week, let’s review continuity, intermediate value theorem, and the limit definition of a derivative.
Continuity
Q1: Classify the discontinuities in the following image:
Q2: Let \(f(x) = \begin{cases} 3x + e^x & \text{if } x \leq 1 \\ x^4 + k & \text{if } x \gt 1\end{cases}\). For what value(s) of \(k\) makes the \(f(x)\) continuous?
Q3: Let \(f(x) = \begin{cases} log(-x) - 1& \text{if } x \leq -1 \\ x & \text{if } -1 \lt x \lt 1 \\ (x - 2)^2 & \text{if } x \gt 1 \end{cases}\). Where is \(f(x)\) continuous and where is it differentiable?
Q4: Let \(f(x) = \begin{cases} \dfrac{(1 - cos(x))(x + 2)}{x^3 + 3x^2 + 5x} & \text{if } x \lt 0 \\ sin(x) & \text{if } x \ge 0 \end{cases}\). Where is \(f(x)\) continuous? Classify the discontinuities.
Q5: Let \(f(x) = \begin{cases} 4 & \text{if } x \lt -1 \\ ax + b & \text{if } -1 \lt x \lt 1 \\ x + 6 & \text{if } x \gt 1 \end{cases}\). What values of \(a\) and \(b\) makes \(f(x)\) have 2 removable discontinuities?
Finding Derivatives using the Limit Definition of a Derivative
Q6: Use the limit definition of a derivative to compute \(f^{'}(x)\) if \(f(x) = x^2 - 5x\).
Q7: Use the limit definition of a derivative to compute \(f^{'}(0)\) if \(f(x) = \dfrac{x^3 - 4x}{x + 3}\).
Q8: Use the limit definition of a derivative to compute \(f^{'}(x)\) if \(f(x) = sin(x)\).
Differentiability using the Limit Definition of a Derivative
Q9: Show that \(f(x) = x^{\tiny \dfrac{2}{3}}\) is not differentiable at \(x = 0\) using the limit definition of a derivative.
Intermediate Value Theorem
Q10: Show that \(f(x) = e^{3x} + 4x^3 + 3\) has a root in \(x \in [-1, 2]\).
Q11: Show that the line \(y = 2\) and the function \(y = 3(x - 4)(x + 2)(x - 3)\) intercept each other on \(x \in [1, 5]\).
