2.5 Homework - Continuity (Homework)

1. [-/1 Points] DETAILS SCALCET9 2.5.011. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The toll 7 charged for driving on a certain stretch of a toll road is $1 except during rush hours (between 7 a.m. and 10 a.m. and between 4 p.m. and 7 p.m. ) when the toll is $3. (a) Sketch a graph of 7 as a function of the time t, measured in hours past midnight. T H 3 3 1 1 - t t - t 7 10 16 19 24 7 10 16 19 24 7 10 16 19 24 7 10 16 19 24 (b) Discuss the discontinuities of of this function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) t = Classify the discontinuities as removable, jump, or infinite. O removable O jump O infinite O none - T is continuous Discuss the significance of the discontinuities of this function to someone who uses the road. O Because of the steady increases and decreases in the toll, drivers may want to avoid the highest rates at t = 7 and t = 24 if feasible. The function is continuous, so there is no significance. Because of the sudden jumps in the toll, drivers may want to avoid the higher rates between t = 0 and t = 7, between t = 10 and t = 16, and between t = 19 and t = 24 if feasible. Because of the sudden jumps in the toll, drivers may want to avoid the higher rates between t = 7 and t = 10 and between t = 16 and t = 19 if feasible.2. [-/1 Points] DETAILS SCALCET9 2.5.015. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. p(v) = 6V 5v2 + 11, a = 1 lim p(V) = lim 6\\ 5v2 + 11 = 6 lim v 5v2 + 11 by the ---Select--- = 6\\ lim (5v2 + 11) by the ---Select--- = 61/ vm (5vz ) + lim 11 by the ---Select-- = 6 5 lim (vz ) + lim 11 v- 1 by the ---Select--- = 6V5 . (1) + 11 by the ---Select--- T..... -Select- Sum Law Difference Constant Multiple Law Find p(1). Product Law Quotient Law P(1) = Power Law Root Law Direct Substitution Property Thus, by the definition of continuity, p is continuous at a = 1. Need Help? Read It Watch It Submit Answer3. [-/1 Points] DETAILS SCALCET9 2.5.019. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Explain why the function is discontinuous at the given number a. (Select all that apply.) H( x ) = - x+ 3 = = -3 [R(-3) and lim f(x) are finite, but are not equal. lim f(x) does not exist. ( lim , A(x) and lim _ f(x) are not finite, and are not equal. A(-3) is undefined. O none of the above Sketch the graph of the function. 2 2 2- -5 -5 -2 -2- -2- -2 O DO Need Help? Read It Watch It Submit Answer4. [-/1 Points] DETAILS SCALCET9 2.5.032. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Explain, using these theorems, why the function is continuous at every number in its domain. f (t ) = e-t- In( 6 + +2) O f(t) is a polynomial function, so it is continuous at every number in its domain. O f(t) is a rational function, so it is continuous at every number in its domain. Of(t) is the product of functions that are continuous on the domain of f(t), so it is continuous at every number in its domain. O f(t) is a root function, so it is continuous at every number in its domain. O f(t) is not continuous at every number in its domain. State the domain. (Enter your answer using interval notation.) Need Help? Read It 5. [-/1 Points] DETAILS SCALCET9 2.5.035.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use continuity to evaluate the limit. lim x\\ 13 - x2 x - 3 Need Help? Read It Watch It Master It