Review: Chapters 1.2 – 2.6

MATH 1441 Calculus 1 Review: Chapters 1.2 – 2.6 1. Determine whether each statement is true or false and explain why. a) The lines 𝑦 = 3𝑥 + 17 and 𝑦 = −3𝑥 + 8 are perpendicular. b) The lines 4𝑥 + 3𝑦 = 8 and 4𝑥 + 𝑦 = 5 are parallel. c) The equation 𝑦 = 3𝑥 + 4 represents the equation of a line with slope 4. d) The line that passes through the points (2, 3) and(2, 5)is a horizontal line. e) The line that passes through the points (4, 6) and(5, 6)is a horizontal line. f) The y-intercept of the line 𝑦 = 8𝑥 + 9 is 9. g) The limit of a function may not be equal to value of the function at that point. h) The limit of a product is the product of the limits when each of the limits exists. i) A rational function is continuous everywhere. j) If a rational function has a polynomial in the denominator of higher degree than the polynomial in the numerator, then the limit at infinity must be equal to zero. 2. Determine the domain and range of the indicated function. 𝑥+5 𝑎) 𝑓(𝑥) = 𝑥 2 −1 𝑏) 𝑔(𝑥) = √1 + 𝑥 𝑥 2 −16 3. Is the function 𝑓(𝑥) = { 𝑥 2 −3𝑥−4 8 5 , 𝑖𝑓 𝑥 ≠ 4 𝑜𝑟 − 1 𝑎𝑡 𝑥 = 4 continuous at 𝑥 = 4? 𝑎𝑡 𝑥 = −1? 4. Find lim 4−3𝑥 . Determine the equations for the vertical and horizontal asymptotes. Find the intercepts. 𝑥→∞ 2𝑥−7 5. Show that 𝑔(𝑥) = 𝑥 2 −3𝑥−4 𝑥−4 has a continuous extension to 𝑥 = 4 and find that extension. Find the limit if it exists. For the following exercises, if the initial substitution of 𝑥 = 𝑎 𝑦𝑖𝑒𝑙𝑑𝑠 𝑡ℎ𝑒 0/0 form, look for ways to simplify the function algebraically or use a table to determine the limit. When necessary, state that the limit does not exist. 6. lim (𝑥 4 − 5𝑥 2 + 2) 𝑥→4 𝑥 4 −8𝑥 7. lim 𝑥→2 𝑥(𝑥 2 −4) 𝑥−5 8. lim 𝑥→5 𝑥 2 −7𝑥+2 5𝑥 3 +8𝑥 2 9. lim 𝑥→0 3𝑥 4 −16𝑥 2 lim 10. 𝑥−1 𝑥→1 √𝑥+3−2 11. lim 𝑥→∞ 12. lim 𝑥→∞ 3𝑥 3 −𝑥+1 𝑥 4 +3 𝑥 2 −8𝑥+6 3𝑥−4𝑥 2 13. Consider the following graph of function 𝑓 for exercises a) to e). a) lim 𝑔(𝑥) b) 𝑥→−1 lim 𝑓(𝑥) c) lim 𝑓(𝑥) 𝑥→−3 d) Is 𝑔 continuous at -1? 𝑥→∞ 14. Determine whether each statement is true or false and explain why. a) The limit of the product is the product of the limits when each of the limits exist. b) If the limit of a function exists at a point, then the function is continuous there. c) A polynomial function is continuous everywhere. 15. Find the average rate of change for the function𝑓(𝑥) = −𝑥 2 + 4𝑥 when 𝑥 varies from -4 to 0. 16. 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 6 from 𝑥 = −1 𝑡𝑜 𝑥 = 3 17. Find the simplified form of the difference quotient for the following functions and then find 𝑓 ′ (𝑥). a) 𝑓(𝑥) = 2𝑥 + 3 b) 𝑓(𝑥) = −4𝑥 2 9 c) 𝑓(𝑥) = 𝑥 18. Find the limit if it exists. a) lim tan(2𝑥) 𝑥→0 tan(𝜋𝑥) b) lim 𝑥→0 sin 5𝑥 3 c) lim 𝑥→0 cos(2𝑥)−1 sin 𝑥 19. Let 𝑓(𝑥) = 𝑥 3 − 𝑥 − 1. Use Intermediate Value theorem to show that 𝑓 has a zero between −1 and 2.