Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{The inverse function of }f(x)=e^{x+2}\text{ is }{f}^{-1}(x)=\ln x+2\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }{f}^{-1}(x)=\ln x-2 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The asymptote of the curve }y=3e^{2x}+5\text{ is the line }y=0\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }y=5 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The solution to }{e}^{2x}=-3\text{ is }x=\frac{1}{2}\ln (-3)\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: There are no real solutions.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The derivative of }{3}^x\text{ is }x{3}^{x-1}\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }{3}^x\ln 3 \\ {x}^n\text{ is a power function (variable base, constant power). }{a}^x\text{ is an exponential function (constant base, variable power). They follow different rules.} \end{gathered}$$
• $$\begin{gathered} \text{What is the result of the integral:} \\ \int e^{ax+b} \\ ,dx \end{gathered}$$
  $\frac{1}{a}{e}^{ax+b}+c$
• $$\begin{gathered} \text{You are asked to sketch }y=\ln (x)\text{. } \\ \text{What are the coordinates of the }x\text{-intercept and the equation of the asymptote?} \end{gathered}$$
  $$\begin{gathered} \text{x-intercept: (1, 0)} \\ \text{Asymptote: }x=0\text{ (the y-axis)} \end{gathered}$$
• $\text{What is the condition for the function }f(x)=\ln (3x-6)\text{ to be defined?}$
  $x>2$
• $\text{How do you solve an equation like }{e}^{2x}-5e^x+6=0\text{?}$
  $\text{Use a substitution }u=e^x\text{ to turn it into a quadratic.}$
• $\text{What is the first step to solve the equation }{e}^{2x+3}=7\text{ to find an exact value for }x\text{?}$
  $\text{Take natural logarithms (}\ln \text{) of both sides.}$
• $$\begin{gathered} \text{You are given data following the model }y=a{b}^x\text{ (where }a\text{ and }b\text{ are constants).} \\ \text{To convert this to a linear graph, what variables should you plot on the axes?} \end{gathered}$$
  $\text{Plot }\log y\text{ against }x$
• $$\begin{gathered} \text{True or False?} \\ \text{If the graph of }{\log }_{10}y\text{ against }x\text{ is a straight line with gradient }m\text{ and intercept }c\text{, then the relationship is }y=10^c\cdot m^x\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Relationship: }y=10^c\cdot (10^m{)}^x\text{ (or }y=a{b}^x\text{ where }a=10^c\text{ and }b=10^m\text{).} \end{gathered}$$
• $\text{When transforming a relationship into linear form }Y=MX+C\text{, if the original equation is }y=a{x}^n\text{, what does the gradient of the log-log plot represent?}$
  $\text{The gradient represents }n\text{.}$
• $$\begin{gathered} \text{You are given data following the model }y=a{x}^n\text{ (where }a\text{ and }n\text{ are constants).} \\ \text{To convert this to a linear graph, what variables should you plot on the axes?} \end{gathered}$$
  $\text{Plot }\log y\text{ against }\log x$