MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 94 C) ) A) 1 2

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1 Chapter Calculus MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average rate of change of the function over the given interval. ) = , [-3, ] A) 30 B) 9 C) 37 7 D) 8 ) ) = -3 -, [5, ] ) A) B) - C) -3 D) - Find the slope of the curve at the given point P and an equation of the tangent line at P. 3) = + 5, P(, 3) A) slope is - 5 ; = B) slope is -39; = ) C) slope is 0 ; = D) slope is 3; = 3 - Use the table to estimate the rate of change of at the specified value of. ) =. ) A) B) C) D) 8

2 Solve the problem. 5) The graph below shows the number of tuberculosis deaths in the United States from 989 to Deaths 5) Year Estimate the average rate of change in tuberculosis deaths from 99 to 99. A) About -0 deaths per ear B) About -500 deaths per ear C) About -0.9 deaths per ear D) About -00 deaths per ear Use the graph to evaluate the limit. ) lim 0 f() ) A) 0 B) does not eist C) D) -

3 7) lim f() 0 7) A) does not eist B) 3 C) -3 D) 0 8) lim f() - 8) A) 3 B) - C) D) - 3 Find the limit. 9) lim ( ) A) does not eist B) 0 C) -8 D) 8 9) 0) lim A) 9 B) 7 C) ±7 D) does not eist 0) Find the limit if it eists. ) lim 3 3( + 7)( - ) A) -3 B) 90 C) 50 D) -90 ) 3

4 ) lim A) -9 B) - C) D) 9 ) Find the limit, if it eists ) lim 0 - ) lim 5) lim h 0 A) - B) C) Does not eist D) 0 + ( - ) A) - B) C) 0 D) Does not eist 3h+ + A) B) C) / D) Does not eist 3) ) 5) ) lim - - A) B) Does not eist C) 0 D) ) ) lim 3-3 8) lim 9) lim h 0 A) 7 B) 3 C) 0 D) Does not eist - - A) - B) C) 0 D) Does not eist ( + h)3-3 h A) 3 B) Does not eist C) 3 + 3h + h D) 0 7) 8) 9) Provide an appropriate response. 0) Let lim 7 f() =. Find lim 7 f(). A) 7 B) C) D) 3. 0) ) Let lim 0 f() = -3 and lim 0 g() = -9. Find lim 0 [f() + g()]. A) - B) 90 C) D) )

5 Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form f( + h) - f() lim occur frequentl in calculus. Evaluate this limit for the given value of and function f. h 0 h ) f() = 3-5, = ) A) B) Does not eist C) 8 D) 9 3) f() = 3 +, = 9 3) A) 3 B) Does not eist C) 3 D) 5 ) f() = +, = 9 A) 8 B) C) 3 D) Does not eist ) Provide an appropriate response. 5) It can be shown that the inequalities - cos hold for all values of 0. 5) Find lim cos if it eists. 0 A) B) C) does not eist D) 0 Find the limit. ) If lim f() = 5, find lim f(). A) -7 B) C) 3 D) Does not eist ) Given the interval (a, b) on the -ais with the point c inside, find the greatest value for δ > 0 such that for all, 0 < - c < δ a < < b. 7) a = -0, b = 0, c = -8 7) A) δ = 8 B) δ = C) δ = D) δ = 5

6 Use the graph to find a δ > 0 such that for all, 0 < - c < δ f() - L < ε. 8) 8) f() = - + c = - L = ε = 0. = NOT TO SCALE A) δ = 0. B) δ = -0. C) δ = D) δ = 0. 9) 9) f() = c = 3 L = 3 = ε = NOT TO SCALE A) δ = 0.8 B) δ = 0.9 C) δ = 0.85 D) δ = 0. A function f(), a point c, the limit of f() as approaches c, and a positive number ε is given. Find a number δ > 0 such that for all, 0 < - c < δ f() - L < ε. 30) f() = 0 + 0, L = 0, c = 3, and ε = ) A) δ = B) δ = 0.00 C) δ = 0.00 D) δ =

7 Find the limit L for the given function f, the point c, and the positive number ε. Then find a number δ > 0 such that, for all, 0 < - c < δ f() - L < ε. 3) f() = , c = -, ε = 0.0 3) + A) L = 0; δ = 0.0 B) L = -8; δ = 0.03 C) L = -7; δ = 0.03 D) L = -; δ = 0.0 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Prove the limit statement 3) lim 3 ( - ) = 7 3) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the graph to estimate the specified limit. 33) Find lim (π/)- f() and lim (π/)+ f() 33) A) ; B) π; π C) π ; π D) ; Determine the limit b sketching an appropriate graph. 3) lim f(), where f() = for < for 3) A) B) -7 C) 5 D) 7

8 35) lim f(), where f() = -+ - < 0, or 0 < 3 = 0 0 < - or > 3 35) A) Does not eist B) -8 C) -0 D) Find the limit. 3) lim ) A) 5 B) 5 C) Does not eist D) 5 37) lim ) A) - B) - 8 C) - 5 D) Does not eist sin Find the limit using lim =. =0 sin 5 38) lim 0 38) A) 5 B) 5 C) D) does not eist sin 5 39) lim 0 sin 39) A) 5 B) 5 C) does not eist D) sin 0) lim 0 A) does not eist B) 0 C) - D) 0) 8

9 sin(sin ) ) lim 0 sin A) - B) does not eist C) 0 D) ) Divide numerator and denominator b the highest power of in the denominator to find the limit. + - ) lim + 5 ) A) B) C) D) 0 3) lim t 9t - 7 t - 3 A) does not eist B) 3 C) 7 D) 9 3) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find a function that satisfies the given conditions and sketch its graph. ) lim ± f() = 0, lim - f() =, lim + f() =. ) ) lim g() = -5, lim g() = 5, lim g() = 5, lim g() = )

10 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. - + (7/) ) lim (/) ) A) - 3 B) 3 C) D) - 7) lim - 0+ A) - B) Does not eist C) D) 0 7) 8) lim 3-8) A) 3 B) C) - D) ) lim A) - B) Does not eist C) 0 D) 9) 50) lim ( ) 50) A) - B) - C) 0 D) 3 5) lim ( ) A) B) C) D) 0 5) 0

11 Answer the question. 5) Does lim (-)+ f() eist? 5) f() = - +, 3, -, -3 +, - < 0 0 < < = < < 3 3 < < d (, -) -5 - t A) Yes B) No Solve the problem. 53) To what new value should f() be changed to remove the discontinuit? +, < f() = 8 = +, > A) - B) C) - D) 5 53) Find the intervals on which the function is continuous. + 5) = A) discontinuous onl when = - or = B) discontinuous onl when = C) discontinuous onl when = or = D) discontinuous onl when = - or = 5) Determine if the given function can be etended to a continuous function at = 0. If so, approimate the etended functionʹs value at = 0 (rounded to four decimal places if necessar). If not, determine whether the function can be continuousl etended from the left or from the right and provide the values of the etended functions at = 0. Otherwise write ʺno continuous etension.ʺ 55) f() = tan 55) A) No continuous etension B) f(0) = C) f(0) = onl from the right D) f(0) = onl from the left Find numbers a and b, or k, so that f is continuous at ever point. 5) f() = 8, a + b, 3, < > 5 A) a =, b = B) a =, b = C) a = 8, b = 3 D) Impossible 5)

12 Find the limit. 57) lim A) - B) - C) 0 D) 57) 58) lim A) - B) - C) 0 D) 58)

13 Answer Ke Testname: LIMITS CAL ) D ) C 3) D ) C 5) D ) B 7) D 8) A 9) D 0) B ) B ) C 3) A ) D 5) C ) D 7) B 8) D 9) A 0) C ) C ) A 3) C ) C 5) D ) A 7) B 8) D 9) B 30) C 3) D 3) Let ε > 0 be given. Choose δ = ε/. Then 0 < - 3 < δ implies that ( - ) - 7 = - 8 = ( - 3) = - 3 < δ = ε Thus, 0 < - 3 < δ implies that ( - ) - 7 < ε 33) A 3) A 35) B 3) A 37) B 38) A 39) B 0) C ) D ) D 3) B 3

14 Answer Ke Testname: LIMITS CAL ) (Answers ma var.) Possible answer: f() = ) (Answers ma var.) Possible answer: f() = 5, > 0-5, < ) A 7) A 8) D 9) C 50) D 5) D 5) A 53) B 5) C 55) B 5) B 57) D 58) B