Calculus Test Questions: Derivatives That Don't Exist

📚 Quick Study Guide

• 🔍 Definition: A derivative of a function at a point exists if the limit of the difference quotient exists at that point.
• 🚧 Discontinuities: If a function $f(x)$ is discontinuous at $x=a$, then $f'(a)$ does not exist.
• 🔪 Sharp Turns/Corners: If $f(x)$ has a sharp turn or corner at $x=a$, then $f'(a)$ does not exist because the left-hand and right-hand limits of the derivative are not equal.
• 📈 Vertical Tangents: If $f(x)$ has a vertical tangent at $x=a$, then $f'(a)$ does not exist because the slope approaches infinity.
• 🎢 Cusps: A cusp is a point where the curve has a sharp point and the tangent line is vertical. The derivative does not exist at a cusp.
• 💡 Key Idea: Check for points where the function is not smooth or continuous to find where derivatives don't exist.

🧪 Practice Quiz

1. Question 1: At which of the following points does the derivative of $f(x) = |x-2|$ not exist?
   1. A) $x = 0$
   2. B) $x = 1$
   3. C) $x = 2$
   4. D) $x = 3$
2. Question 2: Consider the function $f(x) = \begin{cases} x^2 & x < 1 \\ 2x & x \\geq 1 \end{cases}$. Does the derivative $f'(1)$ exist?
   1. A) Yes, $f'(1) = 2$
   2. B) Yes, $f'(1) = 1$
   3. C) No, because the function is not continuous at $x=1$
   4. D) No, because the left and right derivatives are not equal
3. Question 3: For what value(s) of $x$ does the derivative of $f(x) = x^{1/3}$ not exist?
   1. A) $x = 0$
   2. B) $x = 1$
   3. C) $x = -1$
   4. D) The derivative exists for all $x$
4. Question 4: Which of the following conditions is sufficient to conclude that $f'(a)$ does not exist?
   1. A) $f(x)$ is continuous at $x = a$
   2. B) $f(x)$ has a removable discontinuity at $x = a$
   3. C) $f(x)$ has a vertical asymptote at $x = a$
   4. D) $f(x)$ is defined at $x = a$
5. Question 5: At which point does the function $f(x) = \sqrt[3]{x^2}$ have a derivative that does not exist?
   1. A) $x = 1$
   2. B) $x = -1$
   3. C) $x = 0$
   4. D) The derivative exists for all $x$
6. Question 6: If a function has a sharp corner at $x=c$, which of the following is true about its derivative at that point?
   1. A) The derivative exists and is equal to zero.
   2. B) The derivative exists and is positive.
   3. C) The derivative exists and is negative.
   4. D) The derivative does not exist.
7. Question 7: Consider a function $g(x)$ that is not continuous at $x=5$. What can you conclude about $g'(5)$?
   1. A) $g'(5)$ exists and is equal to zero.
   2. B) $g'(5)$ exists.
   3. C) $g'(5)$ does not exist.
   4. D) $g'(5)$ could exist, depending on the function.