Chapter 3: Problem 33
In Exercises \(33-36,\) find the first four derivatives of the function. $$y=x^{4}+x^{3}-2 x^{2}+x-5$$
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Generating the Birthday Probabilities Example 5 of this section concerns the probability that, in a group of \(n\) people, at least two people will share a common birthday. You can generate these probabilities on your calculator for values of \(n\) from 1 to \(365 .\) Step 1: Set the values of \(N\) and \(P\) to zero: Step \(2 :\) Type in this single, multi-step command: Now each time you press the ENTER key, the command will print a new value of \(N(\) the number of people in the room) alongside \(P\) (the probability that at least two of them share a common birthday): If you have some experience with probability, try to answer the following questions without looking at the table: (a) If there are three people in the room, what is the probability that they all have different birthdays? (Assume that there are 365 possible birthdays, all of them equally likely.) (b) If there are three people in the room, what is the probability that at least two of them share a common birthday? (c) Explain how you can use the answer in part (b) to find the probability of a shared birthday when there are four people in the room. (This is how the calculator statement in Step 2 generates the probabilities.) (d) Is it reasonable to assume that all calendar dates are equally likely birthdays? Explain your answer.
Standardized Test Questions You should solve the following problems without using a graphing calculator. True or False The derivative of \(y=2^{x}\) is \(2^{x} .\) Justify your answer.
The Derivative of sin 2\(x\) Graph the function \(y=2 \cos 2 x\) for \(-2 \leq x \leq 3.5 .\) Then, on the same screen, graph $$\quad y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$ for \(h=1.0,0.5,\) and \(0.2 .\) Experiment with other values of \(h,\) including negative values. What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.
Radians vs. Degrees What happens to the derivatives of \(\sin x\) and cos \(x\) if \(x\) is measured in degrees instead of radians? To find out, take the following steps. (a) With your grapher in degree mode, graph \(f(h)=\frac{\sin h}{h}\) and estimate \(\lim _{h \rightarrow 0} f(h) .\) Compare your estimate with \(\pi / 180 .\) Is there any reason to believe the limit should be \(\pi / 180 ?\) (b) With your grapher in degree mode, estimate \(\lim _{h \rightarrow 0} \frac{\cos h-1}{h}\) (c) Now go back to the derivation of the formula for the derivative of sin \(x\) in the text and carry out the steps of the derivation using degree-mode limits. What formula do you obtain for the derivative? (d) Derive the formula for the derivative of cos \(x\) using degree-mode limits. (e) The disadvantages of the degree-mode formulas become apparent as you start taking derivatives of higher order. What are the second and third degree-mode derivatives of \(\sin x\) and \(\cos x\) ?
Multiple Choice Which of the following is \(\frac{d}{d x} \tan ^{-1}(3 x) ?\) \((\mathbf{A})-\frac{3}{1+9 x^{2}} \quad(\mathbf{B})-\frac{1}{1+9 x^{2}} \quad\) (C) \(\frac{1}{1+9 x^{2}}\) \((\mathbf{D}) \frac{3}{1+9 x^{2}} \quad(\mathbf{E}) \frac{3}{\sqrt{1-9 x^{2}}}\)
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