34 - Derivatives of the Trig Functions Consider lim sin...

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Unformatted text preview: Derivatives of the Trig Functions Consider lim sin There are several ways to determine the value of this indeterminate limit One way is to graph both sin and on the same plane: as , how do the graphs compare? a zoomed in graph is below Another way is to graph the function sin MTH 173 section 3.4 notes page 1 A geometric argument shown in the text establishes that cos from which, since lim cos Thm: Then, lim cos lim cos cos sin sin , we use the squeeze thm to obtain the . cos cos lim lim lim sin cos sin cos sin cos lim lim sin lim sin cos , so we have the Thm lim cos MTH 173 section 3.4 notes page 2 cos blue: red: cos Now, the derivatives Thm: sin cos graphs should give a clue: Proof: sin lim sin lim sin lim sin cos red: sin sin blue: cos cos sin sin cos sin lim cos sin lim sin sin sin lim cos lim cos cos lim cos sin cos sin cos MTH 173 section 3.4 notes page 3 Now, what about cos ? xx xxxxxxxx yyyyyyyyyyy zzzzzzzzzzzz wwwww w decreasing increasing decreasing increasing Thm: Then: tan cos cos sin cos sin cos sin sin cos cos sin cos cos sin cos cos sin cos sec Similarly, we can obtain: cot sec csc csc sec tan csc cot differentiate product rule required: differentiate sec tan sec sec tan sin sin cos MTH 173 section 3.4 notes page 4 differentiate tan sec sec sec tan sec sec tan quotient rule required: differentiate sin cos remember that the derivative of the product of 3 functions has 3 terms, each containing the derivative of one of the factors and the other factors unchanged, that is . So sin cos cos cos sin cos at . sin sin is the slope at . cos sin , so and the eqn is sin eqn of tangent to Equation will be Thus cos cos Points on cos sin where tangent is horizontal , be .Thus cos sin sin sin sin or MTH 173 section 3.4 notes page 5 This requires that the slope, i.e., sin sin sin sin sin cos cos graph to check: An equation of vertical motion is seconds. a. b. velocity fn graph and sin cos sin , where is in cm and in cos is red, is blue c. d. When does cos for first time? sin ! sec So we solve and then use that how far from does particle travel? the graph suggests that is largest when result in . " " e. when is speed greatest? the speed is greatest when , so at sec MTH 173 section 3.4 notes page 6 lim sin sin from the graph, this limit is about Since we know sin # lim , we use this fact: # # sin lim sin sin lim sin sin lim lim lim sin MTH 173 section 3.4 notes page 7 ...
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