Quiz 1 - Limits and rate of change
Problem 1
Question
Find \[ \lim_{t\rightarrow 0} \frac{\tan(t)\sec(t)}{3t} \]
Solution
\[ \begin{align*} \lim_{t\rightarrow 0} \frac{\tan(t)\sec(t)}{3t} &= \frac{1}{3}\lim_{t\rightarrow 0} \frac{\sin(t)}{\cos(t)} \times \frac{1}{\cos(t)} \times \frac{1}{t} \\ &= \frac{1}{3} \lim_{t\rightarrow 0} \cancelto{1}{\frac{\sin(t)}{t}} \times \frac{1}{\cos^2(t)} \\ &= \frac{1}{3} \lim_{t\rightarrow 0} \frac{1}{\cos^2(t)} \\ &= \boxed{\frac{1}{3}} \end{align*} \]
Problem 2
Question
For function \(y = f(x) = 3x^2 + 1\):
- Find the rate of change \(\frac{\Delta y}{\Delta x}\)
- Find the average rate of change over intervals \([2,3]\) and \([-1,1]\)
Solution
a.
\[ \begin{align*} \frac{\Delta y}{\Delta x} &= \frac{f(x + \Delta x) - f(x)}{x + \Delta x - x} \\ &= \frac{3(x+\Delta x)^2 + 1 - (3x^2 + 1)}{\Delta x} \\ &= \frac{\cancel{3x^2} + 6x\cancel{(\Delta x)} + 3(\Delta x)^\cancel{2} + \cancel{1}- \cancel{ 3x^2} -\cancel{1} }{\cancel{\Delta x}} \\ &= \boxed{6x + 3(\Delta x)} \end{align*} \]
b.
For \([2,3]\): \[ \begin{align*} \frac{f(x_2) - f(x_1)}{x_2 - x_1} &= \frac{f(3) - f(2)}{3 - 2} \\ &= \frac{3(3)^2 + 1 - (3(2)^2 + 1)}{1} \\ &= 27 - 12 = \boxed{15}\\ \end{align*} \] For \([-1,1]\): \[ \begin{align*} \frac{f(x_2) - f(x_1)}{x_2 - x_1} &= \frac{f(1) - f(-1)}{1 + 1} \\ &= \frac{3(1)^2 + 1 - (3(-1)^2 + 1)}{2} \\ &= \frac{0}{2} = \boxed{0}\\ \end{align*} \]