hw4.pdf - Math 2A Single Variable Calculus Homework...

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Math 2A Single Variable Calculus Homework Questions 44Applications of Derivatives4.1Maximum and Minimum Values1. Find the critical points off(x) =x3+6x2-15x.2. Find the critical points ofh(p) =p-1p2+4.3. Find the critical points ofg(θ) =4θ-tanθ.4. Find the absolute maximum and minimum values off(x) =5+54x-2x3on the interval[0, 4].5. Find the absolute maximum and minimum values off(x) =x3-6x2+5 on the interval[-3, 5].6. Find the absolute maximum and minimum values off(x) =xx2-x+1on the interval[0, 3].7. Find the absolute maximum and minimum values off(t) =3t(8-t)on the interval[0, 8].8. An object with weightWis dragged along a horizontal plane by a force acting along a ropeattached to the object. If the rope makes an angleθwith the plane, then the magnitude of theforce isF=μWμsinθ+cosθ,whereμis a positive constant called thecoefficient of frictionand 0θπ/2. Show thatFisminimized when tanθ=μ.9. Show that 5 is a critical number of the functiong(x) =2+ (x-5)3,but thatgdoes not have a local extreme value atx=5.4.2The Mean Value Theorem1. Letf(x) =tanx. Show thatf(0) =f(π)but that there is no numberxin(0,π)such thatf0(c) =0. Why does this not contradict Rolle’s Theorem?2. Verify that the functionf(x) =x3-3x+2 satisfies the hypotheses of the Mean Value Theoremon the interval[-2, 2], and find all numberscthat satisfy the conclusion of the Mean ValueTheorem.3. Verify that the functionf(x) =1/xsatisfies the hypotheses of the Mean Value Theorem on theinterval[1, 3], and find all numberscthat satisfy the conclusion of the Mean Value Theorem.4. Letf(x) =2- |2x-1|. Show that there is no value ofcsuch thatf(3)-f(0) =f0(c)(3-0).Why does this not contradict the Mean Value Theorem?1
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Chapter 4 / Exercise 5
Essential Calculus: Early Transcendentals
Stewart
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5. Show that the equation 2x-1-sinx=0 has exactly one real root.6. Show that the equationx4+4x+c=0 has at most two real roots.7. Suppose that 3f0(x)5 for all values ofx. Show that 18f(8)-f(2)30.8. Suppose thatfandgare continuous on[a,b]and differentiable on(a,b). Suppose also thatf(a) =g(a)and thatf0(x)<g0(x)fora<x<b. Prove thatf(b)<g(b). (Hint: Apply the MeanValue Theorem to the function h=f-g.)9. At 2 pm a car’s speedometer reads 30 mi/h. Ten minutes later it reads 50 mi/h. Prove that atsome time between 2 and 2.10 pm the acceleration of the car is exactly 120 mi/h2.10. A numberais called afixed pointof a functionfiff(a) =a. Prove that iff0(x)6=1 for all realnumbersxthenfhas at most one fixed point.4.3How Derivatives Affect the Shape of a Graph1. Let

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Essential Calculus: Early Transcendentals
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Chapter 4 / Exercise 5
Essential Calculus: Early Transcendentals
Stewart
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