advertisement

c Math 151 WIR, Spring 2010, Benjamin Aurispa Math 151 Week in Review 6 Sections 3.5, 3.6, & 3.7 1. Differentiate the following functions. (a) f (x) = (5x + cos 3x)−6 (b) f (x) = sin2 4x x6 − tan 5x q (c) f (x) = 3x + p 8 3 − 7x x2 !4 5x2 + sec 2x 1 c Math 151 WIR, Spring 2010, Benjamin Aurispa (d) f (x) = cot2 (x2 − x) + csc(cos(tan 4x)) 2. Given the following table of values, calculate the indicated derivatives. x f f0 g g0 0 2 1 1 −2 3 π 6 π 3 6 −3 2 1 5 2 0 (a) h0 (2) if h(x) = g(f (x)) (b) G0 (1) if G(x) = [f (4x − 3)]3 (c) F 0 (0) if F (x) = f (sin 2x + 1) + cos(g(x2 + 3x)) 2 c Math 151 WIR, Spring 2010, Benjamin Aurispa 5−x 3. Find an equation of the tangent line to the graph of f (x) = √ at the point where x = 1. x2 + 3 4. Find dy dx for the equation 3y 4 − 2x2 y 2 = 7x5 + y 3 c Math 151 WIR, Spring 2010, Benjamin Aurispa 5. Find dy dx for the equation cos(2x − y) + sin 3y = 4xy. 6. Find an equation of the tangent line to the graph of (x2 − 7y 3 )2 = 4y at the point (−3, 1). 4 c Math 151 WIR, Spring 2010, Benjamin Aurispa 7. Show that the curves x2 + y 2 = 2x and x2 + y 2 = 6y are orthogonal. 5 c Math 151 WIR, Spring 2010, Benjamin Aurispa 8. The motion of a particle is given by the vector function r(t) =< −5 cos t, 2 sin t >. (a) Sketch the curve. (b) What is the velocity of the object at the point (0, −2)? (c) In what direction is the particle traveling? 9. Find a unit tangent vector to the curve r(t) =< t3 − 1, 3 − 3t2 > at the point (−2, 0). 6 c Math 151 WIR, Spring 2010, Benjamin Aurispa 10. Find vector and parametric equations for a tangent line to the curve r(t) =< 2t + cos t, 4 sin 2t > at t = π. 11. The position of a thrown water balloon in feet after t seconds is given by r(t) =< 2t, 10t − 4t2 >. (a) What are the velocity and speed of the balloon after 2 seconds? (b) With what speed will the balloon hit the ground? 7 c Math 151 WIR, Spring 2010, Benjamin Aurispa 12. Find the angle of intersection of the curves r1 (t) =< t + 4, t2 − 9 > and r2 (s) =< 5 − s, s2 − 6 >. 8