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c Math 151 WIR, Spring 2010, Benjamin Aurispa Math 151 Exam 1 Night Before Drill 1. (a) Use the limit definition to find the derivative of the function f (x) = (b) Use part (a) to find the equation of the tangent line at x = 3. 1 √ 3x + 7. c Math 151 WIR, Spring 2010, Benjamin Aurispa 2. Use the limit definition to find a vector tangent to the curve r(t) =< 3t2 + 2t, 2t + 1 > at the point (16, 5). 2 c Math 151 WIR, Spring 2010, Benjamin Aurispa 3. The position function of an object in linear motion is given by f (t) = in ft and time is measured in seconds. (a) What is the average velocity of the object from t = 2 to t = 5? (b) What is the velocity of the object at t = 3? 3 t−1 where position is measured t+2 c Math 151 WIR, Spring 2010, Benjamin Aurispa 4. Sketch a graph of the function f (x) = |x2 − 4| and determine where f (x) is not differentiable by examining the graph. 5. Find a unit vector which points in the same direction as the vector with initial point (−5, 2) and terminal point (3, 7). 6. Given vectors a = 4i + 10j and b = −i + 5j, find a vector of length 2 that is orthogonal to a − 4b. 4 c Math 151 WIR, Spring 2010, Benjamin Aurispa 7. A plane heads in the direction N 60◦ E with an airspeed of 300 miles per hour. The wind is blowing S 45◦ E at 20 miles per hour. Find the groundspeed of the plane. 8. Two forces are acting on an object placed at the origin. F1 =< −1, 3 > and F2 =< 5, 2 >. Find the magnitude and direction of the resultant force F. 5 c Math 151 WIR, Spring 2010, Benjamin Aurispa 9. Find the scalar and vector projections of < 2, −9 > onto < 6, 3 >. 10. A port is located on a map at coordinates (2, 4). Two ships leave the port, one headed for an island at (5, 9) and the other headed for an island at (4, −2). Find the angle between the two paths. 11. A force of 15 N is applied horizontally in moving an object up a ramp. The ramp has a horizontal displacement of 8 meters and a vertical displacement of 3 meters. How much work is done by the force? 6 c Math 151 WIR, Spring 2010, Benjamin Aurispa 12. Find a Cartesian equation for the curve r(t) =< −3 sin t, 3 cos t + 1 >. Sketch the curve and describe the direction of motion as t increases. 13. Find parametric equations of the line passing through the points (8, 1) and (4, 5). 14. Find the value(s) of a such that the lines r(t) =< 2 − 5t, 6 + at > and r(s) =< 7 + 6s, −1 + 3as > are perpendicular. 7 c Math 151 WIR, Spring 2010, Benjamin Aurispa 15. Compute the following limits or show why the limit does not exist. 3 1 − x→−2 (x + 2)(x − 1) (x + 2)(x + 1) (a) lim 2x − 3x2 (b) lim √ x→∞ x2 + 4x (c) lim x→−1− x2 (x + 4) x2 − 4x − 5 8 c Math 151 WIR, Spring 2010, Benjamin Aurispa 7 − x2 x→∞ x3 − 3x (d) lim * (e) lim r(t) where r(t) = t→4 (f) lim x4 sin x→0 1 x2 t−4 t2 − 16 √ , 2 t + 5 − 3 2t − 3t − 20 −4 9 + c Math 151 WIR, Spring 2010, Benjamin Aurispa (x − 3)(x + 6) x→3 |x − 3| (g) lim (h) lim x→−∞ √ x2 − 5x + x 10 c Math 151 WIR, Spring 2010, Benjamin Aurispa 16. Find all vertical and horizontal asymptotes of f (x) = 6x2 − 4x − 2 . 7x2 − 8x + 1 17. Show that the equation x4 − 3x2 + x − 3 = −2 has two real solutions. 11 c Math 151 WIR, Spring 2010, Benjamin Aurispa 18. Determine where the following function is discontinuous. f (x) = 5 − x2 3x + 7 5 x2 + 9 6x − 4 if if if if if x ≤ −2 −2<x<1 x=1 1<x<3 x≥3 19. Find the value of m that makes the following function continuous everywhere, if possible. f (x) = 2 mx + 3 5 2x + 2m if x < −1 if x = −1 if x > −1 12