Math151 Sample Problems for the Final Fall 20111. Given vectors ~a = ~ı − 2~,~b =< −2, 3 >. Find(a) a unit vector ~u that has the same direction as 2~b + ~a.(b) angle between ~a and~b(c) comp~b~a, proj~b~a.2. Find the work done by by a force of 20 lb acting in the direction N50oW in moving anobject 4 ft due west.3. Find the distance from the po int (-2,3) to the line 3x − 4y + 5 = 0.4. Find vector and parametric equations for the line passing thro ugh the points A(1, −3)and B(2, 1).5. Find all points of discontinuity for the functionf(x) =x2+ 1 , if x < 2,x + 2 , if x ≥ 2.6. Find the vertical and horizontal asymptotes of the curve y =x2+ 43x2− 3.7. Finddydxfor each function(a) y = (sin x)x.(b) y =5√2x − 1(x2− 4)23√1 + 3x(c) y(t) = sin−1t, x(t) = cos−1(t2).(d) 2x2+ 2xy + y2= x.8. Find the equation of the tangent line to the curve y = x√5 − x at the point (1,2).9. A particle moves on a vertical line so that its coordinate at time t is y = t3− 12t + 3,t ≥ 0.(a) Find the velocity and acceleration functions.(b) When is the particle moving upward?(c) Find the distance that particle travels in the time interval 0 ≤ t ≤ 310. The vector f unction ~r(t) =< t, 25t − 5t2> represents the position of a particle at time t.Find the velocity, speed, and acceleration at t = 1.11. Find y′′if y = e−5xcos 3x12. Findd50dx50cos 2x113. A ladder 10 ft long rests against a vertical wall. If the bot tom of the ladder slides awayfrom the wall at a rate of 0.9 ft/s, how fast is the angle between the ladder and the groundchang ing when the bot tom of the ladder is 8 ft from the wall?14. Find the quadratic approximation of 1/x for x near 4.15. If f(x) = x + x2+ exand g(x) = f−1(x), find g′(1).16. Solve the equation ln(x + 6) + ln(x − 3) = ln 5 + ln 217. Find cos−1sin5π4.18. Evaluate each limit:(a) limx→0sin x + sin 2xsin 3x(b) limx→0(cot x − csc x)(c) limx→0xsin x19. Find the absolute maximum and absolute minimum values of f(x) = x3− 2x2+ x on[-1,1].20. For the function y = x2exfind(a) All asymptotes.(b) Intervals on which the function is increasing/decreasing.(c) All local minima/local maxima.(d) Intervals on which the function is CU/CD.(e) Inflection points.21. A cylindrical can without a top is made to contain V cm3of liquid. F ind the dimensionsthat will minimize the cost of the metal to make the can.22. Find the derivative of the function f ( x) =√xR0t2t2+ 1dt23. Evaluate the integral:(a)2R1x +1x2dx(b)2R1x2+ 1√xdx(c)π/2R0(cos t + 2 sin t)dt24. Find the area under the curve y =√x above the x-axis between 0 and 4 .225. A particle moves in a straight line and has acceleration given by a(t) = t2− t. Its initialvelocity is v(0) = 2 cm/s and its initial displacement is s(0) = 1 cm. Find the positionfunction s(t).26. Find the vector function ~r(t) that gives the position of a particle at time t having theacceleration ~a(t) = 2t~ı + ~, initial velocity ~v(0) = ~ı −~, and initial position (1,
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