c Math 151 WIR, Spring 2013, Benjamin Aurispa Math 151 Week in Review 11 Sections 5.1, 5.2, & 5.3 1. Given the graph of f ′ below, find the following. Then sketch a possible graph of f if f is continuous and f (−6) = 0. (i) The intervals where f is increasing and decreasing. (ii) The x-values of any local extrema of f . (iii) The intervals where f is concave up and concave down. (iv) The x-values of any inflection points of f . 2. Sketch a graph of a function f with the following properties: (i) f is continuous everywhere except at x = 3. (ii) lim f (x) = 0 x→±∞ (iii) lim f (x) = ∞, lim f (x) = −∞ x→3− x→3+ (iv) f (0) = −3, f (7) = 3 (v) f ′ (x) > 0 on (−2, 3) ∪ (3, 7) (vi) f ′ (x) < 0 on (−∞, −2) ∪ (7, ∞) (vii) f ′′ (x) > 0 on (−5, 3) ∪ (5, 7) ∪ (7, ∞) (viii) f ′′ (x) < 0 on (−∞, −5) ∪ (3, 5) 1 c Math 151 WIR, Spring 2013, Benjamin Aurispa 3. Find all absolute and local extrema for the following functions by graphing. (a) f (x) = x2 − 3, −1 ≤ x ≤ 2 (b) f (x) = ( x+2 −(x − 2)2 + 3 if − 2 < x ≤ 0 if 0 < x < 3 4. Find all critical values for the following functions. 1 (a) f (x) = x 3 (x − 1)2 x+5 (b) f (x) = ex (c) f (x) = |x2 − 5x| 5. Find the absolute maximum and minimum values for the following functions on the given interval. (a) f (x) = x4 − 8x2 + 1 on [−1, 3] (b) f (x) = x + 2 sin x on [0, π] √ (c) f (x) = 3 x2 − 6x on [−1, 4] 1 (d) f (x) = on [1, 3] (x − 2)2 6. Sketch a possible graph with the following properties. (a) x = 3 is a critical number of f , but f has no local extrema. (b) f is continuous and has a local minimum at x = 3, but f ′ (3) does not exist. 7. Find the value of c that satisfies the Mean Value Theorem for the function f (x) = 3 − x2 on the interval [1, 6]. 8. An object in motion has initial position s(0) = 2. During the first 8 seconds of motion, the object’s velocity ranges from a minimum of 3 m/s to a maximum of 10 m/s. Use the Mean Value Theorem to give a range of possible positions at time t = 8. 9. Determine all intervals where the function f (x) = x6 − 6x5 is increasing and decreasing, concave up and concave down. Identify all local extrema and points of inflection. 10. Find the intervals where the following functions are increasing/decreasing and identify all local extrema. 2 (a) f (x) = xe−2x x+3 (b) f (x) = (x − 2)2 (c) f (x) = (x − π2 ) cos x − sin x on [0, 2π] 11. Find the intervals where the following functions are concave up and concave down and identify all inflection points. (a) f (x) = ex (x3 − 8x2 + 27x − 38) (b) f (x) = (3 − ln x)3 2 c Math 151 WIR, Spring 2013, Benjamin Aurispa 12. Given the following information, identify any local extrema. x −3 4 6 8 f (x) 1 5 3 4 f ′ (x) 0 0 0 DNE f ′′ (x) −4 0 12 DNE 3