Slopes of tangent lines – practice for tonight

Posted on February 19, 2014 by Kate Poirier

At the end of today’s class each group used the limit formula to compute slopes of tangents for f(x)=\frac{1}{x} for points of tangency with x-coordinates latex 1, 2, 3, 4, and 5. The slopes were -1, -\frac{1}{4}, -\frac{1}{9}, -\frac{1}{16}, and -\frac{1}{25} respectively. From this, you all guessed that the slope of the line tangent to the graph y = f(x) at the point where x=a was -\frac{1}{a^2}. (In class I called it “i” instead of “a” but this isn’t a significant difference.) You were correct and it’ll be almost trivial to see this formally, which is roughly the content of the first part of Section 3.2 and tomorrow’s lesson.

If you’ve got time tonight, it’d be worth practicing a few more examples like this. Some suggested functions are: f(x)=x^2, f(x)=x^3, f(x)=\sqrt{x}, f(x)=|x|. For each of the functions,

  • select a few different points of tangency (a,f(a)) on the graph y=f(x),
  • use the limit definition to compute the slopes of the tangent lines at each of the points you’ve chosen,
  • from these results, guess what the slope of the tangent at (a, f(a)) would be, for general a.

If anyone does this tonight, it’d be great to see your results. You can share them as a comment on this post, or put them up on the board before tomorrow’s class begins. Just make sure you’re clear each time what your choice of f(x) is and what your x=a is.

Just a reminder, given a function f(x) and a point x=a, the slope of the line tangent to the graph y=f(x) at x=a is given by

\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}.

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