Question 1 - Easy (2 points)
Version 1
The slope \(m_{\tan}\) of the tangent line to the curve \(y = g(x)\) at the point \(R(x_0,g(x_0))\) is given by
\(m_{\tan}=\lim_{x\to x_0}\) _________ \(=\lim_{h\to0}\) _________
Fill in the two blanks. You must clearly and coherently write your final answer with the appropriate limit in front of your final answer. Circle your final answers.
Version 2
The slope \(m_{\tan}\) of the tangent line to the curve \(y = k(x)\) at the point \(Q(x_0,k(x_0))\) is given by
\(m_{\tan}=\lim_{x\to x_0}\) _________ \(=\lim_{h\to0}\) _________
Fill in the two blanks. You must clearly and coherently write your final answer with the appropriate limit in front of your final answer. Circle your final answers.
Version 3
The slope \(m_{\tan}\) of the tangent line to the curve \(y = n(x)\) at the point \(U(x_0,n(x_0))\) is given by
\(m_{\tan}=\lim_{x\to x_0}\) _________ \(=\lim_{h\to0}\) _________
Fill in the two blanks. You must clearly and coherently write your final answer with the appropriate limit in front of your final answer. Circle your final answers.
Version 4
The slope \(m_{\tan}\) of the tangent line to the curve \(y = s(x)\) at the point \(T(x_0,s(x_0))\) is given by
\(m_{\tan}=\lim_{x\to x_0}\) _________ \(=\lim_{h\to0}\) _________
Fill in the two blanks. You must clearly and coherently write your final answer with the appropriate limit in front of your final answer. Circle your final answers.
Question 2 - Medium (3 points)
Version 1
Find an equation of the normal line to the graph of \(y = u(x)\) at \(x = −4\), if \(u(−4) = 5\) and \(u'(−4) = −3\). You must clearly and coherently justify your work. You cannot provide only the final answer. Your final answer should be simplified to \(y = mx + b\) form. Circle your final answer.
Version 2
Find an equation of the normal line to the graph of \(y = g(x)\) at \(x = −3\), if \(g(−3) = 11\) and \(g'(−3) = −2\). You must clearly and coherently justify your work. You cannot provide only the final answer. Your final answer should be simplified to \(y = mx + b\) form. Circle your final answer.
Version 3
Find an equation of the normal line to the graph of \(y = k(x)\) at \(x = 4\), if \(k(4) = −5\) and \(k'(4) = 5\). You must clearly and coherently justify your work. You cannot provide only the final answer. Your final answer should be simplified to \(y = mx + b\) form. Circle your final answer.
Version 4
Find an equation of the normal line to the graph of \(y = n(x)\) at \(x = −1\), if \(n(−1) = −3\) and \(n'(−1) = 10\). You must clearly and coherently justify your work. You cannot provide only the final answer. Your final answer should be simplified to \(y = mx + b\) form. Circle your final answer.
Question 3 - Medium (3 points - 1.5 points for each part)
Version 1
Find \(F^{\prime}(\pi)\) given that \(f(\pi)=1\), \(f^{\prime}(\pi)=-2\), \(g(\pi)=2\), and \(g^{\prime}(\pi)=-1\).
Part A: \(\begin{aligned} F(x)=x^2(4f(x)-7g(x)) \end{aligned}\)
Part B: \(F(x)=\frac{xf(x)}{7x+2g(x)}\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for Part A and Part B.
Version 2
Find \(G^{\prime}(\pi)\) given that \(f(\pi)=-2\), \(f^{\prime}(\pi)=1\), \(g(\pi)=-1\), and \(g^{\prime}(\pi)=2\).
Part A: \(G(x)=5x(3f(x)-8g(x))\)
Part B: \(G(x)=\frac{2f(x)}{x^2+2g(x)}\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for Part A and Part B.
Version 3
Find \(H^{\prime}(\pi)\) given that \(s(\pi)=1\), \(s^{\prime}(\pi)=-1\), \(t(\pi)=-2\), and \(t^{\prime}(\pi)=2\).
Part A: \(H(x) = 5x(3t(x) − 8s(x))\)
Part B: \(H(x)=\frac{2s(x)}{x^2+2t(x)}\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for Part A and Part B.
Version 4
Find \(H^{\prime}(\pi)\) given that \(s(\pi)=2\), \(s^{\prime}(\pi)=-2\), \(t(\pi)=-1\), and \(t^{\prime}(\pi)=1\).
Part A: \(H(x) = x 2 (4t(x) − 7s(x))\)
Part B: \(H(x)=\frac{xs(x)}{7x+2t(x)}\)
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer for Part A and Part B.
Question 4 - Difficult (4 points)
Version 1 & 2
Let \(f(x)=\frac{-1}{\sqrt{3-x}}.\) Using the definition of the derivative, find \(f^{\prime}(1)\).
You cannot use differentiation techniques.
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 3 & 4
Let \(f(x)=\frac{1}{\sqrt{4-x}}.\) Using the definition of the derivative, find \(f^{\prime}(2)\).
You cannot use differentiation techniques.
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Question 5 - Challenging (4 points)
Version 1 & 4
Suppose that
\[f(x)=\left\{\begin{array}{ll}5x+2x^4\cos\left(\frac{5}{x}\right)&\quad\text{if}x\ne0\\0&\quad\text{if}x=0\end{array}\right.\]
Is \(f(x)\) differentiable at \(x = 0\)?
Hint: Squeeze theorem.
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.
Version 2 & 3
Suppose that
\[f(x)=\left\{\begin{array}{ll}-3x+2x^4\sin\left(\frac{4}{x^2}\right)&\quad\text{if} x\neq0\\0&\quad\text{if} x=0\end{array}\right.\]
Is \(f(x)\) differentiable at \(x = 0\)?
Hint: Squeeze theorem.
You must clearly and coherently justify your work. You cannot provide only the final answer. Circle your final answer.