# domain and the range

Midterm Exam Review MATH 178 NAME_____________________________ 1. Find the following limits: x2 − x − 2 x →−1 x 2 + 3 x + 2 a. lim 3x + 2 b. lim x→2 3×3 − 3x 2 − 6 x s →−1 x2 + x c. lim 3x 2 − 5 x → 2 x 2 − 4 e. lim d. lim x→4 1 x−4 x3 − 5 x → 2 x 4 − 4 x + 1 f. lim  x2 2. Given the following, sketch the graph and find the limits: f ( x ) =  3 − x a. Find lim− f ( x) x →2 b. Find lim+ f ( x) x →2 c. Find lim f ( x) x →2 if x2 if x2  x2 + 2 3. Given the following: f ( x) =  6 x + 9 if if x  −1 x  −1 a. Find f ( −3) b. Find f (5) c. Find f (−1) 4. Given the following graph, identify the domain and the range. 5. Find the domain for the following functions: f ( x) = 5 − 7 x b. f ( x) = x−4 2 x − 15 x + 7 2 6. Find the slope of the tangent line on the curve f ( x) = 5x 2 − 3x + 8 at the point (−2,34) 7. Find the equation of the tangent line to the curve y = 9 − 2 x 2 at the point (2, 1). Use: y = mx + b or y − y1 = m( x − x1 ) 8. Find the following derivatives: a. f ( x) = 4 x3 − 5x 2 + 8x + 2 b. f ( x) = ( x 2 + 3)( x 2 − 5) c. f ( x) = 5 x7 d. f ( x) = x+2 x −3 e. f ( x) = x2 − 5x − 1 f. g ( x) = ( x 3 − 5 ) 7 g. f ( x) = 3x ln x  x3  h. y = ln    2x − 7  i. f ( x) = 10e4 x − 3×2 j. g ( x) = k. f ( x) = ln(5x 2 − 3x + 6) l.  x +5 g ( x) =    x  2 4×2 ln x 4 9. Given f ( x) = 4 x3 − 2 x 2 − 5x + 3 find f (−2) 10. Find f (x) given f ( x) = 1 ( 2 x − 1) 3 11. A ball is thrown vertically upward with a velocity of 80 ft/sec its height after t seconds is given by the function: s(t ) = −16t 2 + 80t . Solve the following using calculus techniques discussed in class. a. What is the velocity of the ball after 2 seconds? b. What is the maximum height reached by the ball? 12. A company that produces and sells cellular phones determines that the cost of producing x telephones is C ( x) = 0.45x 2 − 15x + 250 dollars. Find the rate at which the cost is changing when 100 cellular phones are produced. 13. Given the following graph, find the limits: a. lim f ( x) = __________ x →1− d. lim− f ( x) = __________ x →2 b. lim+ f ( x) = __________ c. lim f ( x) = __________ e. lim f ( x) = __________ f. lim+ f ( x) = __________ x →1 x →−2 x →1 x →2 14. A bank advertises that it compounds interest continuously and that it will double your money in 15 years. Find the function that satisfies the equation and what is the annual interest rate? Hint: use P = Po e rt 15. How old is a Chinese artifact that has lost 60% of its Carbon-14? Carbon-14 has a decay rate of 0.01205% per year. Hint: use N = N o e − rt 16. The average cost C of a prime-rib dinner was \$4.65 in 1962. Assuming that the exponential growth model applies, and that the exponential growth rate is 5.1%, what will be the cost of such a dinner in 2018? Hint: use P = Po e rt Solutions: 1. 2. 3. 4. 5. a) b) c) d) e) f) 8 -3 –9 DNE (does not exist) 3/2 0 a) 4 b) 1 c) DNE a) 11 b) 39 c) 3 D: (-3, 1] R: [-4, 0] a. (-inf, 5/7] b. All reals except ½ and 7 6. m = -23 7. y= – 8x + 17 8. a) f ( x) = 12 x 2 − 10 x + 8 b) −5 c) f ( x) = −35 x8 d) f ( x) = e) f ( x) = 2x − 5 2 x² − 5x − 1 f) g ( x) = 21x 2 ( x3 − 5 ) 2 3 2 − x x−7 8 x ln x − 4 x j) g ( x) = (ln x)2 i) f ( x) = 80xe4 x − 6 x 2 k) f ( x) = ( x − 3) h) y = g) f ( x) = 3 + 3ln x 9. f ( x) = 4 x ³ − 4 x 10 x − 3 5 x 2 − 3x + 6 l) g ( x) = −20( x + 5)3 x5 f (−2) = 51 10. f ( x) = 48 ( 2 x − 1) 5 11. a) 16 ft/sec b) 100 ft 12. \$75, the cost will increase \$75 when the 101st phone is produced. 13. a) 1 14. 4.6% b) 3 c) DNE d) 3 15. 7604 years e) 4 16. \$80.87 f) 2 6
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