True/False Problems from Old 115 Exam 1 s

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1 Douglass Houghton Workshop, September 2012 True/False Problems from Old 115 Exam 1 s Standard instructions: Answer true only if a statement MUST be true. No explanations are necessary. 1. Winter, 2001 (a) ln(ab) = (ln A)(ln B) (b) lne 2t 1 = 2t 1 (c) sin(3a) = 3 sin(a) (d) As x, x 100 dominates x. (e) log(10a) = loga+1 (A > 0) (f) A fifth degree polynomial must have at least one real zero.

2 2. Fall, 2001 (a) If f(x) is a second-degree polynomial, then f(f(x)) is also a second-degree polynomial. (b) e a+b = e a +e b (c) If f(x) is an exponential function, then f(x) as x. (d) sin(π/3)+x ln(e) +1 is a linear function. (e) The derivative of f(x) at a point is the tangent line at that point. (f) If a is positive, then the function a ln(x) is concave down. (g) If f (x)is anincreasing functiononaninterval, thenf(x) is also increasing on that interval.

3 3. Winter, 2002 (a) The inverse function of g(t) = (1.04) t is g 1 (t) = 1 (1.04) t. (b) ln(2 x +2 x ) = 0. (c) If 22 = 18e 2k, then k = (d) log(67.34(1.03) t ) = t(log(67.34)+log(1.03)). (e) The graph of the function s(t) = 2sin(2t+3) is the graph of the function y = 2sin(2t) shifted 3 units to the left. (f) If f is increasing, then f is increasing.

4 4. Winter, 2004 (a) log ( ) 1 A = log(a). (b) If f(x) = π 5, then f (x) = 5π 4. a (c) The function y = for k > 0 and a,b,c constants b+ce kt has a horizontal asymptote of y = a. c (d) A degree 7 polynomial must have at least 1 real root but cannot have more than 7 real roots. (e) f (a) is the tangent line of f at the point (a,f(a)). (f) If f(x) = x 2, then f 1 (x) = 1 x 2. (g) Iff (a) = 0, thenthepoint(a,f(a))isaninflectionpoint of f.

5 5. Fall, 2004 (a) log 1 (x) = 1 e x. (b) If a function is continuous at a point a, then it must also be differentiable at a. (c) Suppose f is a continuous function on the interval [5,8] and that f(5) = 2 and f(8) = 3. Then f has a zero on the interval (5, 8). (d) lim x 6 x 7 x 7 exists and is equal to 1. (e) Suppose f is a continuous function and f is concave up on the interval ( 10,10). If f (1) = 2, it is possible that f (4) = 3. (f) Supposef isacontinuousfunction,f(1) = 6,andf (x) > 0 for all x between 0 and 5. Then it is possible that f(4) = 6.

6 6. Winter, 2005 (a) Every continuous function is differentiable. (b) If f (x) > 0 for all x in the interval (a,b), then f is increasing on the interval (a, b). (c) By definition, the instantaneous velocity is equal to the difference quotient. (d) Every rational function has a vertical asymptote. (e) If a function is not continuous at a point, then it is not defined at that point. (f) If a function f is decreasing on an interval, then f is decreasing on that interval.

7 7. Fall, 2005 (a) If A and B are positive constants, then the function f(x) = log( Ax+B ) has a vertical asymptote at x = B/A. (b) If an exponential function of t, in years, has decreased to 60% of the original value in two years, in four years it will decrease to 30% of its original value. (c) If h(x) = 1.3(0.5) x then the derivative, h, is decreasing for all x. (d) Thefunctionssin(e x )ande sin(x) areinversesofeachother. (e) If w is a continuous function for all x, then w(x+h) w(x) lim exists for all x. h 0 h (f) If f (x) > 0 onthe interval [a,b], then the average rateof change of f(x) on the interval [a,b] is greater than f (x) for all a < x < b.

8 8. Winter, 2006 (a) If g(x) is an everywhere differentiable function, then so is f(x) = ag(x h)+b, where a, b, and h are constants. (b) Suppose H(t) and T(t) are differentiable functions, and T(t) = H(t) 4. Then H and T have the same derivative at each t. (c) If l and m are inverse functions, and the graph of m crosses the line y = x, the graph of l must also cross this line at the same point. (d) If b is a positive constant, then lim h 0 0.5b 1/2. b+h b h = (e) Ifs(t)givesthepositionofanobjectmovingataconstant velocity, then the object s instantaneous veloctiy at t = a is equal to s(b) s(a) for all a b. b a (f) If t is a differentiable concave up function, then t (a) < t(b) t(a) for all a < b. b a (g) For any constant a, the equation ax = e 2lnx + a 2 has exactly one solution.

9 9. Winter 2007 (a) If f (x) is increasing, then f(x) is also increasing. (b) If f(x) g(x) for all x, then f (x) g (x). (c) There is a function which is continuous on [1,5] but not differentiable at x = 3. (d) Ifafunctionisincreasingonaninterval, thenitisconcave up on that interval.

10 10. Fall 2007 (a) The function f(x) = ex x 2 1 is continuous on [2,5]. (b) Suppose g is a differentiable function on ( 1,1) with g(1) < 0 and g (x) > 0 for x in ( 1,1), then g(x) has a zero on the interval [ 1,1]. (c) If lim x 0 f(x) = lim x 0 +f(x) then f is continuous at x = 0. (d) If x > 0 and e xy 2 = x 2, then y = 2 (1+lnx). x (e) A function that is continuous on [a,b] is always differentiable on [a,b]. (f) If f (a) = 0 and f (x) > 0 for x < a, then f (x) > 0 for x < a.

11 11. Winter 2008 (a) Let f be a non-decreasing differentiable function defined for all x. f (x) 0 for all x. f (x) 0 for all x. f(x) = 0 for some x (b) Let f and g be continuous at x = 1, with f( 1) = 0 and g( 1) = 3. f g is continuous at x = 1. g f is continuous at x = 1. f g is continuous at x = 1. (c) Let f be differentiable at x = 2, with f(2) = 17. limf(x) = 17. x 2 f(2+h) f(2) lim = 17. h 0 h lim h 0 f(2+h) f(2) h exists. (d) Let f be defined on [a,b] and differentiable on (a,b), with f (x) < 0 for all x in (a,b). If a < c < d < b, then f(c) > f(d). f (x) > 0 for some x in (a,b). f is continuous on (a, b). (e) Let f be a twice-differentiable function that is concave-up on (a,b), with f(a) = 4 and f(b) = 1. For some x in (a,b), f(x) = 2.5. For all x in (a,b), f (x) 0. f (a) f (b).

12 12. Fall, 2008 (The following problems are considered to be independent of each other.) (a) Let C(r) represent the total cost of paying off a car loan borrowed at an interest rate of r% per year. Then: The units of C (r) are /year. The expression C (5) = A (with units) represents the rate of change of the total cost of the car loan. The expression C (5) = A (with units) indicates that if the interest rate increases from 5% to 6%, the total cost of the loan would be approximately C(5)+A. The expression C (5) (with units) indicates that if the interest rate increases by 5%, then the total cost of the loan increases by about C (5). The expression C (5) (with units) indicates that if the interest rate increases from 5% to 6%, the total cost of the loan increases by about C (5). The sign of C (5) cannot be determined from the context of the information given. (b) If the figure below shows position as a function of time for two sprinters running in parallel lanes, then: At time A, both sprinters have the same velocity. Both sprinters continually increase their velocity. Both sprinters run at the same velocity at some time before A. At some time before A, both sprinters have the same acceleration. f(t) A t

13 (c) Let f and g be differentiable functions. Assume f is an even function and g is an odd function. Then: g is an even function. the composition, f(g(x)), is an odd function. h(x) = f(x)g(x) is an odd function. (d) Suppose that f (x) > 0 everywhere. Then: f (x) is increasing. f(b) > f(a) whenever a < b. f (x) < 0.

14 13. Fall 2009 (The three parts (a) (c) are independent of each other.) (a) Suppose f is an increasing differentiable function with domain (, ) so that f(1) = 1 and f( 1) = 1. f is linear. There is a number c so that f(c) = 0. lim x 1 f(c) = 1. lim x f(x) =. f (1) 0. (b) Suppose g(t) is the mass (in grams) of mold on a wedge of cheese in a refrigerator t days after it was abandoned. This mass grows exponentially as a function of time for two weeks, when it is finally thrown away. The graph of g is concave up. The continuous growth rate of g is less than the daily growth rate. The amount of time it takes for the mass of the mold on the cheese to triple is 1.5 times the amount of time it takes for it to double. (c) If f(x) = g(x) and h(3) = 0 then h(x) The graph of f has a vertical asymptote at x = 3. 3 is not in the domain of f. f is not continuous on [ 2,2]. lim x 3 f(x) does not exist.

15 14. Winter 2010 (The four parts (a) (d) are independent of each other.) (a) Suppose f is a differentiable function which is concave up on its entire domain, (, ). lim x 1 f(x) = f(1). f(2) f(1) f (2) f (1) (b) Suppose that h(t) gives the height of a ball, measured in feet above the ground level, t seconds after it is thrown off a bridge. Assume that the derivative of h is given by the formula h (t) = 32t+64. The ball reaches its maximum height 2 seconds after being thrown. The ball reaches a maximum height of 64 feet from the ground. The bridge is 64 feet off the ground. (c) Suppose that A and B are positive constants and A < B. (lne A )(lne B ) = A+B. ln(10 A ) < 0. ln(a 2 +B) = 2lnA+lnB. loga < logb. (d) Suppose that f(x) = Ae Bx for some positive constants A and B. f (x) > 0 for all x. f is increasing. f is increasing.

16 15. Winter, In the following problem, circle all of the statements that could be true anddraw a line throughall of the statements that could not be true, based onthe given information. (Every statement should be either circled or crossed out.) No explanation is necessary. (a) A brief table of values for f(x) and g(x) is given, rounded to 4 decimal places: x f(x) g(x) f(x) is exponential and g(x) is a power function. f(x) is a power function and g(x) is exponential. f(x) and g(x) are both exponential. (b) Suppose that f(x) is a continuous function and lim x f(x) = 2. For all x > 10, f (x) > 0 and f (x) > 0. For all x > 10, f (x) > 0 and f (x) < 0. For all x > 10, f (x) < 0 and f (x) > 0. For all x > 10, f (x) < 0 and f (x) < 0. (c) A rational function r(x) is graphed below (with A B): A 1 (d) Consider the functions f(x), g(x), and h(x) graphed below. B x 2 r(x) = (x+a)(x B) x r(x) = (x+a)(x B) 4x 2 r(x) = (x+a)(x B) x 2 r(x) = (x A)(x+B) g f h f(x) = h (x) h(x) = f (x) h(x) = g (x) f(x) = g (x)

17 Answers Exam a) b) c) d) e) f) g) Winter 2001 F T F F T T Fall 2001 F F F T F T F Winter 2002 F F F F F F Winter 2004 T F F T F F F Fall 2004 F F T T F F Winter 2005 F T F F F F Fall 2005 T F F F F F Winter 2006 T T T T T T F Winter 2007 F F T F Fall 2007 T F F T F F Winter 2008 TFF TFT TFT TFT TTT Fall 2008 FFTFTF FFTF TFT TFF Fall 2009 FTTFT TTF FTFF Winter 2010 TFT TFF FTFT TFT Winter 2012 FTF FTTF TFFF FTTT