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If π(π₯) is a differentiable and π(π₯) = π₯ π(π₯) use the definition of the derivative to show
that πβ²(π₯) = π₯π'(π₯) + π(π₯).
Find the domain
π(π₯) = ββπ₯ + 4
π₯
β π₯
π₯.
(b) Fully discuss the continuity of π (π₯) at π = 4, mention any case of
one sided continuity.
Evaluate the following limits, if they exist, where βπ₯β is the greatest integer function.
(a)lim β2π₯β/π₯
π₯β0
(b) lim π₯ β1/π₯β
π₯β0
Let π(π₯) = βπ₯β + ββπ₯β, where βπ₯β is the greatest integer less than or equal to π₯.
(π) For what values of π, does limπ₯βπ
π(π₯)exist?
(π) At what numbers is π discontinuous?
A candy factory produces the candy boxes which include 100 parts in each box. According to the quality control section, 95% of candies are qualified. (a)What is the probability that more than 2 candies in a given box will be non qualified? (b) What is the probability that more than 10 candies in a given box will be non qualified?
In a beauty salon, the waiting time to receive a haircut service is a random variable with a continuous uniform distribution. A calling for a new haircut service is every 10 minutes. (a) What is the probability that a customer waits for a new call more than 7 minutes? (b) What is the probability that a customer waits for a new call between 2 and 7 minutes?
Β In Istanbul, on average, 6 Leveche (lodos) occur every year. (a) What is the probability that in a given year, number of lodos is fewer than 4 in Istanbul? (b) What is the probbaility that in a given year, number of lodos is from 6 to 8 in Istanbul?Β
ACTIVITY IN BASIC CALCULUS
BASIC RULES IN DERIVATIVE
Β Complete the blanks of the given function below with a number (except 0 and 1) to create your own problem and find the derivative of the function. Show your complete solution to each problem.
Β
1. f(x) = -4x5+ ______x-4Β - 2468
2. f (x) =____x-3- _____x1/4 - 12x
3.f(x)= ____Β "\\sqrt[4]{x^3} - \\underset{x^6}{=} + \\frac{2}{3} x^6"
4.f (x) =Β "\\underset{x^-6}{=} -" ____ x2 + "\\sqrt[4]{x}"
Bacteria grow in a nutrient solution at a rate proportional to the amount present. Initially, there are 250 strands of the bacteria in the solution which grows to 800 strands after seven hours. Find (a) an expression for the approximate number of strands in the culture at any time t and (b) the time needed for the bacteria to grow to 1600 strands.Β
Nick is given $50 to spend on a vacation. He decides to spend $5 a day. The amount Nick has left and the number of days are related. How many days Nick have after $25?
