1.
Question 1 Evaluate \(\lim_{x\to 0}\frac{\cos 3x}{x}\).
Solution: According to the provided solution key, this question's answer is 0. Answer: 0
2.
Question 2 Find the value of \(k\) that will make the function \(f(x)\) continuous everywhere: \(f(x)=\begin{cases}2x+k, & x\le 2\\ kx^2-5, & x>2\end{cases}\).
Solution: For the function to be continuous at \(x=2\), the two pieces must have the same value there. Set \(2x+k=kx^2-5\) at \(x=2\). Then \(2(2)+k=k(2)^2-5\), so \(4+k=4k-5\). Thus \(3k=9\), and \(k=3\). Answer: 3
3.
Question 3 \(f(x)=\begin{cases}x^2-2, & x\le 3\\ x+4, & x>3\end{cases}\) is continuous everywhere.
Solution: Both pieces are continuous on their own intervals, so check the boundary point \(x=3\). The left value is \(f(3)=3^2-2=7\). The right-hand limit is \(\lim_{x\to 3^+}(x+4)=3+4=7\). Since these are equal, the function is continuous at \(x=3\), and therefore it is continuous everywhere. Answer: True
4.
Question 4 Where is \(f(x)=\frac{x-3}{x^2+3x-18}\) discontinuous?
Solution: A rational function is discontinuous where its denominator is zero. Set \(x^2+3x-18=0\). Factor: \((x+6)(x-3)=0\). Therefore, \(x=-6\) or \(x=3\). Answer: -6 and 3
5.
Question 5 What is the instantaneous rate of change of \(y=\frac{1}{x}\) at \(x=2\)?
Solution: The instantaneous rate of change is the derivative. For \(y=\frac{1}{x}=x^{-1}\), \(y'=-x^{-2}=-\frac{1}{x^2}\). At \(x=2\), \(y'=-\frac{1}{2^2}=-\frac{1}{4}\). Answer: \(-\frac{1}{4}\)
6.
Question 6 What is the instantaneous rate of change of \(y=\sqrt{x}\) at \(x=2\)?
Solution: The instantaneous rate of change is the derivative. For \(y=\sqrt{x}=x^{1/2}\), \(y'=\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}\). At \(x=2\), \(y'=\frac{1}{2\sqrt{2}}\). Answer: \(\frac{1}{2\sqrt{2}}\)
7.
Question 7 The position of a particle on the x-axis is given by \(x(t)=t^2-4t+3\). At what time is the particle's velocity zero?
Solution: Velocity is the derivative of position. Since \(x(t)=t^2-4t+3\), \(v(t)=x'(t)=2t-4\). Set the velocity equal to zero: \(2t-4=0\). Then \(2t=4\), so \(t=2\). Answer: \(t=2\)
1 out of 1
