Cards In This Deck

• $\text{What defines a point of inflection in terms of the behavior of the gradient before and after the stationary point?}$
  $$\begin{gathered} \text{The gradient has the same sign on both sides of the stationary point.} \\ \text{(e.g., Positive -> Zero -> Positive OR Negative -> Zero -> Negative).} \end{gathered}$$
• $\text{What is the condition for a stationary point at }x=a\text{ to be a local maximum using the second derivative test?}$
  $f^{\prime }(a)=0\text{ AND }{f}^{\prime \prime }(a)<0$
• $\text{To find the coordinates of a stationary point on }y=f(x)\text{, what are the two essential calculation steps?}$
  $$\begin{gathered} \text{1. Solve }{f}^{\prime }(x)=0\text{ to find the x-coordinate(s).} \\ \text{2. Substitute this x-value back into the original }f(x)\text{ to find the y-coordinate.} \end{gathered}$$
• $$\begin{gathered} \text{You are determining the nature of a stationary point at }x=a\text{ using the second derivative, and you find that:} \\ {f}^{\prime \prime }(a)=0 \\ \text{What is the specific conclusion and the next step?} \end{gathered}$$
  $$\begin{gathered} \text{The test is inconclusive (it could be a maximum, minimum, or point of inflection).} \\ \text{Next Step: You must use the Tabular Method (checking the sign of the gradient }{f}^{\prime }(x)\text{ slightly to the left and right of }x=a\text{).} \end{gathered}$$
• $\text{What is the standard procedure to find the range of values for which }f(x)=x^3+6x^2-15x\text{ is a decreasing function?}$
  $$\begin{gathered} \text{1. Differentiate to find }{f}^{\prime }(x)\text{.} \\ \text{2. Set up the inequality }{f}^{\prime }(x)\le 0\text{.} \\ \text{3. Solve the resulting quadratic inequality (finding critical values and sketching the quadratic).} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{To prove that a function }f(x)\text{ is increasing on an interval }[a,b]\text{, you must show that }{f}^{\prime }(x)>0\text{ for all }x\text{ in that interval.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Condition: }{f}^{\prime }(x)\ge 0 \\ \text{The definition allows the gradient to be zero at discrete points (e.g., a point of inflection) while still being classified as an increasing function.} \end{gathered}$$
• $$\begin{gathered} \text{You are sketching the gradient function }y=f^{\prime }(x)\text{ based on the graph of }y=f(x)\text{.} \\ \text{If the original graph }y=f(x)\text{ has a local minimum at }x=a\text{, what feature must the gradient graph }y=f^{\prime }(x)\text{ show at }x=a\text{?} \end{gathered}$$
  $\text{The graph of }y=f^{\prime }(x)\text{ must cut the x-axis at }x=a\text{ going from negative to positive (below to above axis).}$
• $$\begin{gathered} \text{True or False?} \\ \text{If the curve }y=f(x)\text{ has a horizontal asymptote at }y=5\text{, then the gradient function }y=f^{\prime }(x)\text{ will also have a horizontal asymptote at }y=5\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: The gradient function }y=f^{\prime }(x)\text{ will have a horizontal asymptote at }y=0\text{.} \\ \text{Since the curve flattens out (gradient approaches zero) as }x\to \infty \text{, the value of }{f}^{\prime }(x)\text{ approaches 0.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If }y=f(x)\text{ has a point of inflection at }x=a\text{, then the graph of the gradient function }y=f^{\prime }(x)\text{ cuts through the x-axis at }x=a\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: The graph of }y=f^{\prime }(x)\text{ touches the x-axis (is tangent to it) at }x=a\text{.} \\ \text{The gradient goes from Positive -> 0 -> Positive (or Neg->0->Neg), meaning }{f}^{\prime }(x)\text{ does not cross from positive to negative values.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{When solving an optimization problem (e.g., finding the maximum volume of a tank), finding the value of }x\text{ where }\frac{dV}{dx}=0\text{ is the final step.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Step: You must justify that this value gives a maximum (or minimum) as requested.} \\ \text{Usually, this involves finding }\frac{d^{2}V}{d{x}^2}\text{ and showing it is negative for a maximum (or positive for a minimum).} \end{gathered}$$
• $\text{When finding the minimum surface area of a shape where the total volume is fixed (e.g., }V=1000\text{), what is the immediate first step before differentiating?}$
  $$\begin{gathered} \text{Express one variable in terms of the other using the fixed constraint (e.g., }h=\frac{1000}{ \\ \pi r^2}\text{) and substitute it into the Surface Area formula.} \end{gathered}$$