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c Math 151 WIR, Fall 2013, Benjamin Aurispa Math 151 Exam 1 Week in Review 1. Use the limit definition to find the derivative of f (x) = 3x2 + 2x − 1. 2. (a) Use the limit definition to find the derivative of the function f (x) = √ 3x + 7. (b) Find an equation of the tangent line to the graph of f (x) at the point where x = 3. 3. The position function of an object in linear motion is given by f (t) = in ft and time is measured in seconds. t−1 where position is measured t+2 (a) What is the average velocity of the object from t = 2 to t = 5 seconds? (b) What is the velocity of the object at t = 3 seconds? 4. Consider the function f (x) = |x2 − 2x − 3|. (a) Sketch a graph to determine where f (x) is not differentiable. (b) What can be said about f ′ (0)? f ′ (2)? 5. Compute the following limits or show why the limit does not exist. (a) lim x→−2 3 1 − (x + 2)(x − 1) (x + 2)(x + 1) x2 (x + 4) x→−1− x2 − 4x − 5 7 − x2 (c) lim 3 x→∞ x − 3x 2x − 3x2 (d) lim √ x→∞ x2 + 4x (b) lim (e) lim r(t) where r(t) = t→4 * 25 − x2 x→5+ x2 − 10x + 25 (x − 3)(x − 6) (g) lim x→3 |x − 3| √ (h) lim x2 − 5x + x t2 − 16 t−4 √ , t + 5 − 3 2t2 − 3t − 20 + (f) lim x→−∞ 6. Find all vertical and horizontal asymptotes of f (x) = 6x2 − 4x − 2 . 7x2 − 8x + 1 7. Show that the equation x4 − 3x2 + x − 3 = −2 has a real solution. 8. Determine where the following function is not continuous. f (x) = 5 − x2 3x + 7 5 x2 + 9 6x − 4 x−4 if if if if x ≤ −2 −2 < x< 1 x=1 1<x<3 if x ≥ 3 1 c Math 151 WIR, Fall 2013, Benjamin Aurispa 9. Find the values of m and c that make the following function continuous everywhere. f (x) = 2 mx + 3 c 2x + 2m if x < −1 if x = −1 if x > −1 10. Suppose a is a vector with initial point (−1, 2) and terminal point (3, 12) and that b = −i + 5j, find a unit vector that is orthogonal to a − 4b. 11. 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 (true speed) of the plane. 12. Two forces are acting on an object placed at the origin. F1 =< −2, 3 > and F2 pulls vertically down with a magnitude of 10 pounds. Find the resultant force F and its direction from the positive x-axis. 13. Find the scalar and vector projections of the vector b = 2i − 9j onto the vector a = 6i + 3j. 14. 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). Use vectors to find the angle between the two paths. 15. A force of 15 N is applied horizontally in moving an object 8 meters up a ramp. If the ramp is inclined at a 30◦ angle, how much work is done by the force? 16. Find Cartesian equations for the following curves and sketch a graph. √ (a) x = − t − 4, y = t + 7 (b) r(θ) =< −3 sin θ, 3 cos θ + 1 > 17. Find parametric equations of the line passing through the points (8, 1) and (4, 5). 18. Find the value(s) of a such that the lines r1 (t) =< 2 − 5t, 6 + at > and r2 (s) =< 7 + 6s, −1 + 3as > are perpendicular. 2