Show all work in details for each question. Points: Q1-40. = 4; Q,-4; Q 4: Qs =
ID: 3145121 • Letter: S
Question
Show all work in details for each question. Points: Q1-40. = 4; Q,-4; Q 4: Qs = 10. 1. (1) Give your definition for the Intermediate Value Theorem. (ii) Assume that f(x) -2-cos z + inside the interval [-2,1]. Show there exists at least one solution in the given domain. 2. Use the precise definition of limit to show that the limit of the function is 2+3-4 5 3. (i) Give your definition of the Squeeze Theorem. (ii) What is the following limit? 37T 4. () Give your definition of continuity at a point 'x-a'. (ii) Compute the rate of change at a given point at x-a' and its tangent y=,0(3,1) (0) lim 2 (ii) Inn( ) equation at 'x=a. 5. Compute the following limits: (iii) (z- (tu)lino eosin-1Explanation / Answer
1.) (i) Intermediate value theorem states that if a f is a continuous function with an interval, [a, b], as its domain, and it takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval
(ii) f(x) = 2-cos(x)+x3 in intervel [-2,1]
therefore, f(-2)= 2-cos(-2)-8 = -6-cos(2) (which is negative)
and, f(1)= 2-cos(1)+1 = 3-cos(1) (is positive and belongs in the range (2,3))
Then according to Intermediate value theorem, f(x) must be be between -6-cos(2) and 3-cos(1) in intervel (-2,1)
Let's take f(0), f(0)=2-cos(0)+0= 2-1=1 which surely lies in between -6-cos(2) and 3-cos(1).
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