c Math 151 WIR, Spring 2013, Benjamin Aurispa Math 151 Week in Review 6 Sections 3.4-3.7 1. Calculate the following limits. (a) lim sin 9x x(cos x + 1) (b) lim (cos x − 1) sin 5x x2 (c) lim cot 3x csc 4x (d) lim tan2 3x 6x2 x→0 x→0 x→0 x→0 1 c Math 151 WIR, Spring 2013, Benjamin Aurispa 2. Find the tangent line to the graph of f (x) = tan x + 4 at x = π4 . 3. For what values of x does f (x) = sin x − cos x have a horizontal tangent line in the interval [0, 2π]. 4. Differentiate the following functions. (a) f (x) = sec x cot x + x csc x (b) g(x) = x − cos x tan x + sin x 2 c Math 151 WIR, Spring 2013, Benjamin Aurispa (c) f (x) = (x3 + 1)6 + p 4x2 + 8 + sin(5x3 ) − tan 5x (d) f (x) = 5x + cos 3x + (e) f (x) = r 1 3x2 −6 3 √ 3 + sec 2x x (f) f (x) = sin2 4x + cot4 (x2 − x) (g) f (x) = csc(cos(tan 4x)) 3 c Math 151 WIR, Spring 2013, Benjamin Aurispa 5. Given the following table of values, calculate the indicated derivatives. x f (x) f ′ (x) g(x) g ′ (x) 0 2 1 1 −2 3 π 6 π 3 6 −3 2 1 5 2 0 (a) h′ (2) if h(x) = g(f (x)) (b) G′ (1) if G(x) = [f (4x − 4)]3 (c) Find F ′ (x) if F (x) = cos(g(x2 + 3x)) 5−x 6. Find an equation of the tangent line to the graph of f (x) = √ at the point where x = 1. x2 + 3 4 c Math 151 WIR, Spring 2013, Benjamin Aurispa 7. Find dy dx for the equation 3y 4 − 2x2 y 2 = 7x5 + y 8. Find dy dx for the equation cos(2x − y) + sin 3y = 4xy. 5 c Math 151 WIR, Spring 2013, Benjamin Aurispa 9. Find the slope the tangent line to the graph of (x2 − 7y 3 )2 = 4y at the point (−3, 1). 10. Show that the curves x2 + y 2 = 2x and x2 + y 2 = 6y are orthogonal at the point 6 9 3 5, 5 c Math 151 WIR, Spring 2013, Benjamin Aurispa 11. Find a unit tangent vector to the curve r(t) =< t3 − 1, 3 − 3t2 > at the point (−2, 0). 12. Find vector and parametric equations for a tangent line to the curve r(t) =< 2t + cos t, 4 sin 2t > at t = π. 7 c Math 151 WIR, Spring 2013, Benjamin Aurispa 13. 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? 14. Find the angle of intersection of the curves r1 (t) =< t + 4, t2 − 9 > and r2 (s) =< 5 − s, s2 − 6 >. 8