28 - Defn: The derivative of a function at a number ,...

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Unformatted text preview: Defn: The derivative of a function at a number , denoted by lim , is if this limit exists. Alternate fom of definition of derivative: lim 20. The limit lim and . to see that , determine represents the derivative of some function at some number . Determine Compare with lim 16. Given , so we see that and here we have , so, substituting gives lim lim lim lim lim lim lim lim lim MTH 173 section 2.8 notes page 1 Recall that the slope of the line tangent to the graph of lim lim at is given by Now we can say that the slope of this tangent line is given by the derivative of 12. Let a. tan . Estimate the value of in two ways: at . By using the definition of derivative and evaulating for values of very close to . tan tan I do this with a table, the first column being ,the second column being values of As , it appears that b. By zooming in on the graph of tan and estimating the slope. MTH 173 section 2.8 notes page 2 this window is from to here the points on this little section of the graph are about We approximate the derivative by calculating the slope between these points: and 4. If the tangent line to at passes thru the point , determine and Since is the point of tangency, it lies on the graph on , so that . Since the tsngent line passes thru and , and since the derivative gives slope of tangent line, we have that 6. slope of tangent line and Sketch the graph of a function for which A possible graph MTH 173 section 2.8 notes page 3 8. If , determine at the point and use it to determine an equation of the tangent line to the curve Here I'm using the alternative form of the derivative: lim lim lim is lim Thus the equation of the line with slope and containing graph on next page 26. A particle moves along a line with an equation of motion and in seconds. Determine the velocity when . Recall that velocity lim lim lim lim is lim lim lim m/s , where is measured in meters , which is the derivative of . MTH 173 section 2.8 notes page 4 Since lim lim gives average rate of change of gives the (or of ) over the interval and since instantaneous rate of change of or at , and since these expressions are the same as the derivative, we have a general interpretation of the derivative as giving the instantaneous rate at which is changing with respect to at 28. a. The number of bacteria after hours in a controlled lab experiment is What is the meaning of ?. What are its units? This is the rate of growth of the population at hour . In other words, the rate at which the population is increasing. b. Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, or ? is probably larger because there are probably more bacteria to reproduce. If the supply of nutrients was limited, would that affect your conclusion? If the supply were limited, at som e time the population can't get any bigger, and possibly would even decrease. If the population was at a maximum when , then since there would be a 0 growth rate. MTH 173 section 2.8 notes page 5 ...
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