Math 151 WIR, Spring 2010,cBenjamin AurispaMath 151 Exam 2 Review1. Use differentials to approximate the value of4√16.2.2. Find the linear approximation to f(x) = (x − 2)3at x = 4 and use it to approximate the valueof 2.053.1Math 151 WIR, Spring 2010,cBenjamin Aurispa3. Find the quadratic approximation to f(x) = cos x at x =π3.4. Calculate the following limits.(a) limx→6+12x−8x−6(b) limx→∞e5x− e−2xe6x− e−4x2Math 151 WIR, Spring 2010,cBenjamin Aurispa5. Find an equation of the tangent line to the graph of f(x) = xex5+xat x = 1.6. Consider the function f(x) =x3− 4x3− 9.(a) Show that f(x) is one-to-one.(b) Find the inverse of f(x).3Math 151 WIR, Spring 2010,cBenjamin Aurispa7. Given f(x) = e3x+3− 4x3− 2, find g0(3) where g is the inverse of f.8. A 5-meter drawbridge is raised so that the angle of elevation changes at a rate of 0.1 rad/s. At whatrate is the height of the drawbridge changing when it is 2 m off the ground?9. A dog sees a squirrel at the base of a tree 10 ft away. The dog takes off running toward the tree witha speed of 3 ft/s. The squirrel takes off up the tree at the same time with a speed of 5 ft/s. At whatrate is the distance between them changing 2 seconds later?4Math 151 WIR, Spring 2010,cBenjamin Aurispa10. A 25-ft long trough has ends which are isosceles triangles with height 5 ft and a length of 4 ft acrossthe top. Water is poured in at a rate of 15 ft3/min, but water is also leaking out of the trough at arate of 3 ft3/min. At what rate is the water level changing when the width of the water across thetrough is 2 ft?11. Calculate the following limits.(a) limx→0cos x + tan 9x − 1sin x(b) limx→0x cos 7x sin 4x3 sin210x5Math 151 WIR, Spring 2010,cBenjamin Aurispa12. Find the values of x, 0 ≤ x ≤ 2π where the tangent line to f(x) = sin2x + cos x is horizontal.13. Find y0for the equation sin(3x − y) + xy2= ex3y.14. Find the slope of the tangent line to the graph of 5x2y − y3= −12 at the point (1, 3).6Math 151 WIR, Spring 2010,cBenjamin Aurispa15. For what value(s) of a are the curvesx2a2+y29=54and y2− 4x2= 5 orthogonal at the point (1, 3)?16. Find a tangent vector to the curve r(t) =< cos33t, 2 sin 5t + cos 4t > at the point where t =π4.7Math 151 WIR, Spring 2010,cBenjamin Aurispa17. The position of an object is given by the function r(t) =<√t2+ 8t,t3t − 2>.(a) What is the speed of the particle at time t = 1?(b) What is the acceleration at time t = 1?18. Find f(29)(x) where f(x) = e−4x+ cos 3x.8Math 151 WIR, Spring 2010,cBenjamin Aurispa19. Consider the curve x = t2+ 6t, y = 2t3− 9t2.(a) Find the slope of the tangent line at the point (−5, −11).(b) Find the points on the curve where the tangent line is horizontal or vertical.20. Find an equation of the tangent line to the curve x = tan 2t +4(t + 1)2, y = (t2+ 3t + 2)4− (t + 1)5/2at the point where t = 0.9Math 151 WIR, Spring 2010,cBenjamin Aurispa21. Suppose F (x) =3pg(7x + 1) + f (5x + g(cos x)) +g(f (3x)). Use the table of values below to find F0(0).x 0 1 2f 2 1 7f05 6 3g 2 1 −3g0−4 24 −122. Show that the function f (x) = e2xcos x satisfies the differential equation y00− 4y0+ 5y = 0.10Math 151 WIR, Spring 2010,cBenjamin Aurispa23. Differentiate the following.(a) f(x) = tan25x + sec(√3x2− x3)(b) g(t) = 3t2+ 5t4t3− t2!8(c) h(x) = (6x2−4√8x6− x)5(cot 9x +
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