Motivating Questions
How do we put together all the thing Calculus tells us about a function to make a sketch of its graph?
How do we put together all the thing Calculus tells us about a function to make a sketch of its graph?
Let \(f(x)=\frac{1}{3}x^3-\frac{1}{2}x^2-2x+9\).
The three main steps in curve sketching are the following:
Let \(\displaystyle f(x)=\frac{1}{x^2-9}\).
After doing all this work, you may be wonderiing why we did not just plug the function into a calculator or computer to graph. While technology can be very useful, there are a few reasons to do these by hand.
The computational cell below gives an in-between option. It computes some things that you would need for analysis by hand.
The order in which you gather information about the function is not so important, but it is useful to have a list so you do not forget anything.
Determine the domain of \(f\). Look for places to exclude, such as division by 0, even roots of negative numbers, logarithms of 0 or negative numbers, etc.
Compute \(f(0)\) to get the point \((0,f(0))\) on the graph.
Set \(f(x)=0\) and see if you can solve it. Sometime this is too hard (or impossible) and can be skipped.
If \(f(x)=f(-x)\) (such as with \(f(x)=x^2\)) then the function is even. You can save half the work by sketching \(f\) for \(x \ge 0\) and then reflecting it over the \(y\)-axis to get the sketch of \(f\) for \(x \le 0\). It is okay to skip this step, but it can save you work.
If \(f(x)=-f(-x)\) (such as with \(f(x)=x^3\)) then the function is odd. You can save half the work by sketching \(f\) for \(x \ge 0\) and then reflecting it over the \(y\)-axis and then over the \(x\)-axis to get the sketch of \(f\) for \(x \le 0\). It is okay to skip this step, but it can save you work.
If \(f(x+p)=f(x)\) for some \(p\not= 0\) (such as with \(f(x)=\sin(x)\) for \(p=2\pi\)) then the function is periodic. You can then do the work and sketch on the interval \([0,p]\) and then say it continues periodically.
If \(\displaystyle \lim_{x\rightarrow a^+} f(x)= \infty\), \(\displaystyle \lim_{x\rightarrow a^-} f(x)=\infty\), \(\displaystyle \lim_{x\rightarrow a^+} f(x)=- \infty\), or \(\displaystyle \lim_{x\rightarrow a^-} f(x)=-\infty\), then \(f\) has a vertical asymptote at \(x=a\). This normally happens when you would divide by 0 at \(a\) or take the logarithm of 0 at \(a\).
If \(\displaystyle \lim_{x\rightarrow \infty} f(x)=L\) or \(\displaystyle \lim_{x\rightarrow -\infty} f(x)=L\) (with \(L\not=\pm\infty\)), then \(f\) has a horizontal asymptote at \(y=L\).
Compute \(f'(x)\). Find when \(f'(x)\) equals zero or does not exist to locate the critical numbers. Make a sign chart to determine the intervals where \(f\) is increasing and the intervals where \(f\) is decreasing.
Use your sign chart to determine which critical numbers give maximums, which give minimums, and which give neither. Compute the values of the maximums and minimums by plugging the critical number \(c\) into \(f\). This gives a point \((c,f(c))\).
Compute \(f''(x)\). (You may want to simplify \(f'(x)\) first.) Find when \(f''(x)\) equals zero or does not exist to locate potential changes in concavity. Make a sign chart to determine the intervals where \(f\) is concave up and the intervals where \(f\) is concave down.
Use your sign chart for \(f''\) to determine when the concavity changes. Compute the values of \(f\) at these locations to get inflection points \((c,f(c))\).
Analyze and graph the function \(f(x)=x\sqrt{5-x}\).
Analyze and graph the function \(f(x)=e^{-x^2}\).
Analyze and graph the function \(f(\theta)=2\cos(\theta)+\cos^2(\theta)\).
Analyze and graph the function \(f(x)=x\ln(x)\).