Chapter 3: Problem 54
True or False The graph of \(f(x)=1 / x\) has no horizontal tangents. Justify your answer.
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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 Find \(y^{\prime \prime}\) if \(y=x \sin x\) (A) \(-x \sin x\) (B) \(x \cos x+\sin x\) (C) \(-x \sin x+2 \cos x\) (D) \(x \sin x\) (E) \(-\sin x+\cos x\)
Particle Motion A particle moves along a line so that its position at any time \(t \geq 0\) is given by the function \(s(t)=\) \(t^{3}-6 t^{2}+8 t+2\) where \(s\) is measured in meters and \(t\) is measured in seconds. (a) Find the instantaneous velocity at any time t. (b) Find the acceleration of the particle at any time t. (c) When is the particle at rest? (d) Describe the motion of the particle. At what values of t does the particle change directions?
Finding Profit The monthly profit (in thousands of dollars) of a software company is given by \(P(x)=\frac{10}{1+50 \cdot 2^{5-0.1 x}}\) where x is the number of software packages sold. (a) Graph \(P(x)\) (b) What values of \(x\) make sense in the problem situation? (c) Use NDER to graph \(P^{\prime}(x) .\) For what values of \(x\) is \(P\) relatively sensitive to changes in \(x\) ? (d) What is the profit when the marginal profit is greatest? (e) What is the marginal profit when 50 units are sold 100 units, 125 units, 150 units, 175 units, and 300 units? (f) What is \(\lim _{x \rightarrow \infty} P(x) ?\) What is the maximum profit possible? (g) Writing to Learn Is there a practical explanation to the maximum profit answer?
Group Activity In Exercises \(43-48,\) use the technique of logarithmic differentiation to find \(d y / d x\) . $$y=x^{\tan x}, x>0$$
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