Practice Test (Midterm 1) 1. Evaluate the following limits (if exist) and also find the horizontal asymptotes for the last two problems (if exist). 5x4 − 4x3 + 2 . x→+∞ 1 − 3x2 − x4 x2 − 1 (d) lim . x→−∞ x (x2 − 2x)(x2 − 4) x→2 (x − 2)2 2 3x + 1 (b) lim 3 x→−1 x + x2 (c) (a) lim lim 2. Find the points of discontinuity of the following functions: 3 x x+1 2 if x < −1 3 − 4x + 1 (b) g(x) = 4 (x − 4) (a) f (x) = 2x2 − 1 if − 1 ≤ x ≤ 0 3 3x + x2 − x − 1 if x > 0 3. If lim f (x) = 1, then find lim [f (x) + x2 − 1]. x→4 x→4 5 , then what is the value of f (c)? 3 5. Find the slope and the instantaneous rate of change of the following functions at any point x and the specified points: 4 2x − 1 1 (a) y = ). , at (1, 16 x2 + x √ 3 2x − 1 (b) f (x) = , at (0, −1). 2x + 1 6. Find the slope of the horizontal tangents of the following curves: 4. Let f be a function which is continuous at x = c, if lim f (x) = x→c (a) f (x) = x6 − 6x4 + 8. p (b) y = x2 − x + 4. (c) for a function f (x) for which f 0 (x) = x(x − 1) . x2 + 1 7. Find (a) y (4) , if y = x6 − 15x3 . √ d2 y (b) y (5) , if = 3 3x + 2. 2 dx x2 (3) (c) f (x), if f 0 (x) = 2 . x +1 √ 8. If f (x) = x + x, find the rate of change of f (4) at x = 1. 9. Suppose a particle is moving along a straight line, at time t seconds its distance from the starting point is s meter. If s is related to t by the following formula: s = 100 + 10t + 0.01t3 Find it’s velocity and acceleration at time t = 2 seconds. 10. Solve the problems 27 and 28 from Page No. 675 of your text book. 11. Solve the problems 18, 20, 27, 30, 33 − 36 from Page No. 703 of your text book.