MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points

Transcription

1 MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to lim f (x) 0 2. lim f (x) DNE x x 5 3. lim x 1 f (x). lim x 1 + f (x) 5. f (2) 3 6. f ( 3) 0 7. Name te type of discontinuity tat exists in f wen x = 1. infinite discontinuity 8. Name one point on te grap of f at wic te function is continuous but is not differentiable. State te value of x were tis occurs and explain wy it is bot continuous and not differentiable. At x = 1 At x = 1, tere is a corner in te grap of f. At x = 1, te derivative from te left is not equal to te derivative from te rigt, so it is impossible for a derivative to exist at x = 1. Te function continuous at x = 1, because te left- and rigt-side limits are te same and are bot equal to te value of te function (f(1)=3) at tat pint. Use te grap of a function sown ere as you respond to questions 9 and Arrange te following values in order from least to greatest: (2.5),(1) ( 1),(0), (2). Ordered: (0), (2),(2.5),(1) ( 1) 10. On te interval 3 x 3, state te x-value intervals on wic (x) < 0. Solution : (x) < 0 for 2 < x <1 11. State te first initial and last name for eac of te two individuals wo are credited wit creating calculus more tan 300 years ago. I. Newton and G. Leibniz

2 If a ball is trown into te air wit a velocity of 0 ft/s, its eigt (in feet) after t seconds is given by (t) = 0t 16t 2. Use tis information for questions12 troug 1 below. Please carefully ceck te units of measure you attac to your responses! 12. In tis context, wat is te meaning of (1) = 2? Here, (1) = 2 means tat at time t = 1 second, te eigt of te ball is 2 feet. 13. Explain te meaning of for tis situation. Here, (1) = 8 means tat at time t = 1 second, te rate of cange of te eigt of te ball te instantaneous rate of cange, or instantaneous velocity is 8 feet per second. 1. Suppose you were told tat at some instant in time, (t) = 0. Wat does tat mean in tis situation? Here, (t) = 0 means tat at some time t seconds into te fligt of te ball, its instantaneous velocity is 0 feet per second, meaning te ball is not moving upward nor downward at tat instant. For tis situation, tis is te time t at wic te ball canges direction from upward movement to downward movement. x Suppose you enter te following function into your calculator: y1(x) =. If you x 2 + 3x ten enter te following command on your ome screen, limit(y1(x),x,1), te calculator reports tat te specified limit as a value of undef. (a) Circle one coice: Has te calculator reported te correct limit? YES NO Tat isn t te best response to te limit question. Rater tan being undefined, we say te limit does not exist (DNE). In tis case te limit is undefined because te two one-sided limits are not te same. (b) Justify your response to (a) by means oter tan troug te use of te calculator s limit command. (i) Look at te grap of y1(x) and note tat te two one-sided limits are not te same. For x approacing 1 from te left, y1(x) approaces negative infinity. As x approaces 1 from te rigt, y1(x) approaces positive infinity. (ii) Take a numerical approac. Let x get arbitrarily close to 1 from te left (0.9, 0.99, 0.999, , and so on) and watc te function values grow negatively witout bound. Let x get arbitrarily close to 1 from te rigt (1.1, 1.01, 1.001, , and so on) and watc te function values grow positively witout bound. (iii) Ply te S & S Wat-If Game! After factoring te rational function, we ave y1(x) = (x+2)/[(x-1)(x+)]. As x gets close to 1, wat appens to te function?

3 16. Use te definition of derivative to determine g (x) for g(x) = x. To earn any credit for x +1 your solution, you must sow appropriate algebraic evidence leading from te definition of derivative for tis function to a final simplified derivative. For g(x) = x x +1 : g(x + ) g(x) g (x) (x +) (x + )+1 x x +1 ( x + )(x +1) (x + +1)(x +1) (x)(x + +1) (x +1)(x + +1) x 2 + x + x + (x + +1)(x +1) x 2 + x + x (x +1)(x + +1) x 2 + x + x + x 2 x x (x + +1)(x +1) (x + +1)(x +1) 1 (x + +1)(x +1) (x + +1)(x +1) = (x +1)(x +1) = g (x) (x +1) 2 = 17. Use your answer for question #16 to write an equation of te tangent line to g(x) at x=3. Sow evidence for your calculations. g'(x) = and (x +1) g (3) = 2 16 = 1 g(x) = x x +1 g(3) = 12 = 3 So te tangent line equation wit m = 1 troug te point (3,3) is y 3 = 1 (x 3) y = 1 x Determine te average rate of cange for te function f ( x) = x 2 x + 3 on te interval [0,]. Average rate of cange is just te 2-point slope of te function for te interval specified. Here, x 1 = 0 and x 2 =, wit y 1 = f(0) = 3 and y 2 = f() = 3. Because y 1 = y 2, we know te 2-point slope is 0.

4 19. Dan said tat e tougt te function in te previous question, f ( x) = x 2 x + 3, ad a value of 1 for some x-value on te interval [0,2]. Sow and explain ow te Intermediate Value Teorem may be used to justify tat for some x value on te interval [0,2]. Te Intermediate Value Teorem (IVT) tells us tat for a function f tat is continuous on a specified closed interval [a,b], te function f is assured to take on any value between f(a) and f(b). More specifically, assuming tat f(a) < f(b) [it works te oter direction as well], te IVT assures tat for any value k, f(a) < k < f(b), tere exists at least one value c, a < c < b, suc tat f(c) = k. So for te specific function provided ere, f(0) = 3 and f(2) = 1. Because f is a polynomial, it is continuous over any closed interval. Terefore, f meets te conditions of te IVT. Because of tat, we can be certain tat tere exists at least one value x = c, 0 < c < 2, suc tat f(c) = Use te coordinate axes below to sketc a function y = f(x) tat as all of te following properties: Te function as a orizontal asymptote of y = 1 as x. Te function as a orizontal asymptote of y =-1 as x.

5

6 Calculus I MAT 15 Test #2 Evaluation Criteria Part I (Only Part): Calculators Allowed 100 points (1) troug (7): 3 points eac (8) troug (10): 5 points eac (11): points (12) troug (1): 5 points eac (15): (a) 3 points (b) 7 points (16): 10 points (17): 5 points (18): 5 points (19): 5 points (20): 10 points