11.14 Applications of the differentiation of exponential functions
Interactive practice questions
Consider the function $f\left(x\right)=3-e^{-x}$f(x)=3−e−x.
Determine $f\left(0\right)$f(0).
Determine $f'\left(0\right)$f′(0).
Which of the following statements is true?
$f'\left(x\right)<0$f′(x)<0 for $x\ge0$x≥0
$f'\left(x\right)<0$f′(x)<0 for all real $x$x.
$f'\left(x\right)>0$f′(x)>0 for all real $x$x.
$f\left(x\right)>0$f(x)>0 for all real $x$x.
Determine the value of $\lim_{x\to\infty}f\left(x\right)$limx→∞f(x).
Find the equation of the tangent to the curve $f\left(x\right)=2e^x$f(x)=2ex at the point where it crosses the $y$y-axis.
Express the equation in the form $y=mx+c$y=mx+c.
Consider the curve $f\left(x\right)=e^x+ex$f(x)=ex+ex.
By filling in the gaps, complete the proof showing that the tangent to the curve at the point $\left(1,2e\right)$(1,2e) passes through the origin.
Consider the function $f\left(x\right)=4e^{-x^2}$f(x)=4e−x2.