Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
Find the derivatives of the following
a) \(y=e^{f\left(x\right)} \)
b) \(y=\left(f\left(x\right)\right)^{n} \)
c) \(y=\cos\left(f\left(x\right)\right) \)
d) \(y=\ln\left(f\left(x\right)\right) \)
Solutions
Get Ready
Questions
a) By considering the graph of \(y=\sin x\), determine $$\lim_{x\to 0} \sin x$$How could we determine this algebraically?
b) By considering the graph of \(y= x\), determine $$\lim_{x\to 0} x$$How could we determine this algebraically?
Now consider $$\lim_{x\to 0}\frac{\sin x}{x}$$
c) Why can we not just evaluate the function at the limit?
d) Using your answers to parts (a) and (b) what is the problem with trying to determine the value of this limit algebraically?
Now consider a different example:
e) By considering the graph of \(y=x^{2}\), determine $$\lim_{x\to \infty} x^{2}$$How could we determine this algebraically?
f) By considering the graph of \(y=e^{-x}\), determine $$\lim_{x\to \infty} e^{-x}$$How could we determine this algebraically?
Now consider $$\lim_{x\to \infty}x^{2}e^{-x}$$
g) Why can we not just evaluate the function at the limit?
h) Using your answers to parts (e) and (f) what is the problem with trying to determine the value of this limit algebraically?
Solutions
Notes
In certain functions it is not obvious whether a limit exists, and if it does what the value is.
There are 3 indeterminate forms that we will consider.
\(\lim\frac{f(x)}{g(x)}\) where both \(\lim f(x)=0\) and \(\lim g(x)=0\). This results in the indeterminate form \(\frac{0}{0}\).
\(\lim\frac{f(x)}{g(x)}\) where both \(\lim f(x)=\pm\infty\) and \(\lim g(x)=\pm\infty\). This results in the indeterminate form \(\frac{\pm\infty}{\pm\infty}\).
\(\lim f(x)g(x)\) where \(\lim f(x)=0\) and \(\lim g(x)=\pm\infty\). This results in the indeterminate form \(0\times\pm\infty\).
In each of these cases, the indeterminate form cannot be evaluated as is, because the function is not defined at the point.
l’Hopital’s Rule gives us a way to determine limits of the first two types. It states that, if $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{0}{0}\text{ OR }\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\pm\infty}{\pm\infty}$$where \(a\) is either real or \(\pm\infty\), then$$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f’(x)}{g’(x)}$$That is, the limit of the indeterminate form is equal to the limit of the derivative of the numerator over the derivative of the denominator.
For limits of the form \(\lim_{x\to a}f(x)g(x)=0\times\pm\infty\), you can always rearrange the function into one of the other two forms.
Note that the formula for l’Hopital’s Rule is NOT in the formula booklet.
Examples and Your Turns
Example
Use l’Hopital’s Rule to evaluate$$\lim_{x\to 0}\frac{2^{x}-1}{x}$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{t\to 1}\frac{5t^{4}-4t^{2}-1}{10-t-9t^{3}}$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{x\to \infty}\frac{\ln x}{x}$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{x\to \infty}\frac{e^{x}}{x^{2}}$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{x\to \infty}xe^{-x}$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{x\to \infty}x\ln x$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{x\to \infty}xe^{x}$$
Your Turn
Use l’Hopital’s Rule to evaluate$$\lim_{x\to \infty}\frac{\ln \left(\cos 3x\right)}{\left(\cos 2x\right)}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.