Covers Chapter 1 Sample Quiz Number 1, Math 1251, Scot Adams, Fall 1998 No calculators. 50 minutes. Every problem is worth one point. Answers: DDACGD*TBGFAGFJTCICICITCHDDBFG %***************** 1. The equation of the tangent line to y=2x^2 at the point (2,8) is (A) y-2=8(x-4) (B) y+8=2(x-2) (C) y=x (D) y-8=8(x-2) (E) y-8=4x(x-2) (F) none of these %section 1.7 Tangents, velocities and other rates of change %***************** 2. Which of these lines is a horizontal asymptote for f(x)= { 150x^2-2x+4 \over 10x^2+150x+1000 }? (A) x=2 (B) y=2 (C) y=x (D) y=15 (E) x=10 (F) x=15 %section 1.6 Limits at infinity; horizontal asymptotes %***************** 3. Let L be the tangent line to y=x^2+3x-11 at the point (2,-1). Then the x-intercept of L is (A) 2-(-1/7) (B) 2-(1/7) (C) -2+(1/7) (D) -2+(-1/7) (E) 1-(-2/7) (F) 1-(2/7) (G) -1+(2/7) (H) -1+(-2/7) (I) 7-(-1/2) (J) 7-(1/2) (K) -7+(1/2) (L) -7+(-1/2) (M) none of these %section Preview of calculus %***************** 4. Compute \lim_{h\to0} {(12+h)^5-12^5 \over h}. (A) 12^5 (B) 4(12^5) (C) 5(12^4) (D) does not exist (E) h^5/h (F) 12^5+5(12^4)h+10(12^3)h^2+10(12^2)h^3+5(12)h^4+h^5 (G) 5(12^4)h+10(12^3)h^2+10(12^2)h^3+5(12)h^4+h^5\over h (H) none of these %section 1.3 Calculating limits using the limit laws %***************** 5. Estimate 12.002^5, using the fact that 12.002^5-12^5\over0.002} is very close to \lim_{h\to0} {(12+h)^5-12^5 \over h}. (A) 10(12^5) (B) 4(12^5) (C) 5(12^4) (D) h^5/h (E) 12^5+12^5 (F) 12^5+4(12^5) (G) 12^5+5(12^4)(0.002) (H) 12^5+h^5\over h (I) 12^5(0.002)+12^5 (J) 12^5(0.002)+4(12^5) (K) 12^5(0.002)+5(12^4) (L) 12^5(0.002)+h^5\over h (M) none of these %section 1.2 The limit of a function %***************** 6. Write down an intuitive definition of \lim_{x\to\infty} f(x)=-\infty. (A) If x very negative, then f(x) is very positive. (B) If x is very positive, then f(x) is very positive. (C) If x is very negative, then f(x) is very negative. (D) If x is very positive, then f(x) is very negative. (E) If x very close to 0, then f(x) is very negative. (F) If x very close to 0, but not equal to 0, then f(x) is very negative. (G) If x very positive, then f(x) is very close to 0. (H) If x very positive, then f(x) is very close to 0, but not equal to 0. (I) none of these %section 1.4 The precise definition of a limit %***************** 7. Compute (d/dx)(\sin(\cos(x))) %section 1.4 The precise definition of a limit %***************** 8. True or False: There are functions which have two horizontal asymptotes. (T) True (F) False %section 1.2 The limit of a function %***************** 9. Compute \lim_{x\to 3^-} {4 \over x-3}. (A) \infty (B) -\infty (C) 0 (D) 4/3 (E) 3/4 (F) does not exist (G) none of these %section 1.2 The limit of a function %***************** 10. Compute \lim_{x\to2} {1+x \over 1-x}. (A) 0 (B) 1 (C) -1 (D) 2 (E) -2 (F) 3 (G) -3 (H) does not exist (I) none of these %section 1.5 Continuity %***************** 11. True or False: Every rational function is continuous at every real number. (T) True (F) False %section 1.5 Continuity %***************** 12. Compute \lim_{x\to1} {x^2+2x-3 \over x^2+x-2}. (a) 4/3 (b) 3/4 (c) 2/3 (d) Does not exist %section 1.3 Calculating limits using the limit laws %***************** 13. Compute \lim_{h\to0} {\sqrt{16+h}-\sqrt{16} \over (16+h)-16}. (A) \sqrt{16} (B) 1/\sqrt{16} (C) 0 (D) 16 (E) 1/16 (F) 1/32 (G) 1/(2\sqrt{16}) (H) 1-\sqrt{16} (I) does not exist (J) none of these %section 1.3 Calculating limits using the limit laws %***************** 14. True or False: Suppose \lim_{x\to2^+}f(x)=\infty and \lim_{x\to2^-}f(x)=-\infty. Then y=2 is a horizontal asymptote for f. (T) True (F) False %section 1.6 Limits at infinity; horizontal asymptotes %***************** 15. Compute \lim_{x\to1} {x-1 \over {\sqrt{3x-2}-1}}. (A) 3/4 (B) 4/5 (C) 5/6 (D) 6/7 (E) 7/8 (F) 8/9 (G) 9/10 (H) 10/11 (I) 11/12 (J) none of these %section 1.3 Calculating limits using the limit laws %***************** 16. True or False: The Intermediate Value Theorem implies that x^5+3x^2+4 has a root between -100 and 100. (T) True (F) False %section 1.5 Continuity %***************** 17. Suppose that a particle is moving along a real number line, and suppose that its position at time t is t^2+3t. Find the average velocity between times t=1 and t=3. (A) (3^2+3\cdot3)-3 \over 3^2+3\cdot1)-1 (B) 3(3-1)^2+3\cdot(3-1) \over 3-1} (C) (3^2+3\cdot3)-(1^2+3\cdot1) \over 3-1 (D) (3^2+3\cdot3)-(1^2+3\cdot1) (E) 3-1 (F) (3^2+3\cdot3)^3-(1^2+3\cdot1)^3 \over 3^3-1^3 (G) (3^2+3\cdot3)^2-(1^2+3\cdot1)^2 \over 3^2-1^2 (H) 1^2+3\cdot1 (I) 2\cdot1+3 (J) 2\cdot1 (K) 2\cdot2+3 (L) none of these %section 1.7 Tangents, velocities, and other rates of change %***************** 18. Suppose that a particle is moving along a real number line, and suppose that its position at time t is t^2+3t. Find the instantaneous velocity at time t=1. (A) (3^2+3\cdot3)-3 \over 3^2+3\cdot1)-1 (B) 3(3-1)^2+3\cdot(3-1) \over 3-1 (C) (3^2+3\cdot3)-(1^2+3\cdot1) \over 3-1 (D) (3^2+3\cdot3)-(1^2+3\cdot1) (E) 3-1 (F) (3^2+3\cdot3)^3-(1^2+3\cdot1)^3 \over 3^3-1^3 (G) (3^2+3\cdot3)^2-(1^2+3\cdot1)^2 \over 3^2-1^2 (H) 1^2+3\cdot1 (I) 2\cdot1+3 (J) 2\cdot1 (K) 2\cdot2+3 (L) none of these %section 1.7 Tangents, velocities, and other rates of change %***************** 19. The amount of water in a tank t minutes after we start our stopwatch is t^2+3t gallons. Find the average rate (in gallons per minute) at which water is entering the tank between times t=1 minute and t=3 minutes. (A) (3^2+3\cdot3)-3 \over 3^2+3\cdot1)-1 (B) 3(3-1)^2+3\cdot(3-1) \over 3-1 (C) (3^2+3\cdot3)-(1^2+3\cdot1) \over 3-1 (D) (3^2+3\cdot3)-(1^2+3\cdot1) (E) 3-1 (F) (3^2+3\cdot3)^3-(1^2+3\cdot1)^3 \over 3^3-1^3 (G) (3^2+3\cdot3)^2-(1^2+3\cdot1)^2 \over 3^2-1^2 (H) 1^2+3\cdot1 (I) 2\cdot1+3 (J) 2\cdot1 (K) 2\cdot2+3 (L) none of these %section 1.7 Tangents, velocities, and other rates of change %***************** 20. The amount of water in a tank t minutes after we start our stopwatch is t^2+3t gallons. Find the instantaneous rate (in gallons per minute) at which water is entering the tank at time t=1 minute. (A) (3^2+3\cdot3)-3 \over 3^2+3\cdot1)-1 (B) 3(3-1)^2+3\cdot(3-1) \over 3-1 (C) (3^2+3\cdot3)-(1^2+3\cdot1) \over 3-1 (D) (3^2+3\cdot3)-(1^2+3\cdot1) (E) 3-1 (F) (3^2+3\cdot3)^3-(1^2+3\cdot1)^3 \over 3^3-1^3 (G) (3^2+3\cdot3)^2-(1^2+3\cdot1)^2 \over 3^2-1^2 (H) 1^2+3\cdot1 (I) 2\cdot1+3 (J) 2\cdot1 (K) 2\cdot2+3 (L) none of these %section 1.7 Tangents, velocities, and other rates of change %***************** 21. Imagine the graph of y=f(x)=x^2+3x. Find the slope of the secant line between (1,f(1)) and (3,f(3)). (A) (3^2+3\cdot3)-3 \over 3^2+3\cdot1)-1 (B) 3(3-1)^2+3\cdot(3-1) \over 3-1 (C) (3^2+3\cdot3)-(1^2+3\cdot1) \over 3-1 (D) (3^2+3\cdot3)-(1^2+3\cdot1) (E) 3-1 (F) (3^2+3\cdot3)^3-(1^2+3\cdot1)^3 \over 3^3-1^3 (G) (3^2+3\cdot3)^2-(1^2+3\cdot1)^2 \over 3^2-1^2 (H) 1^2+3\cdot1 (I) 2\cdot1+3 (J) 2\cdot1 (K) 2\cdot2+3 (L) none of these %section 1.7 Tangents, velocities, and other rates of change %***************** 22. Imagine the graph of y=f(x)=x^2+3x. Find the slope of the tangent line to this graph at (1,f(1)). (A) (3^2+3\cdot3)-3 \over 3^2+3\cdot1)-1 (B) 3(3-1)^2+3\cdot(3-1) \over 3-1 (C) (3^2+3\cdot3)-(1^2+3\cdot1) \over 3-1 (D) (3^2+3\cdot3)-(1^2+3\cdot1) (E) 3-1 (F) (3^2+3\cdot3)^3-(1^2+3\cdot1)^3 \over 3^3-1^3 (G) (3^2+3\cdot3)^2-(1^2+3\cdot1)^2 \over 3^2-1^2 (H) 1^2+3\cdot1 (I) 2\cdot1+3 (J) 2\cdot1 (K) 2\cdot2+3 (L) none of these %section 1.7 Tangents, velocities, and other rates of change %***************** 23. True or False: Let f(x)=|x|. Then f is continuous at every real number. (T) True (F) False %section 1.5 Continuity %***************** Note: {\it only} means that the word "only" should be in italics. 24. Determine the numbers where f(x)= 3x+1, if x\le2 x^2+3, if 2