Homework Answers

Suppose that f:[0,2]→ R is continuous on [0,2] and differentiable on [0,2] and

that f(0) =0 , f(1) =1, f(2) =1.

(i) Show that there exists c↓1∈ (0,1)such that f'(c↓1) =1

(ii) Show that there exists c↓2 ∈ (1,2)such that f'(c↓2) =0.

(iii) Show that there exists c ∈ (0,2)such that f'(c) =1/3

Find the integrating factor and solve the following equations:

( 𝑦 − 𝑥^2) 𝑑𝑥 +( 𝑥^2sin 𝑦 − 𝑥 )𝑑𝑦 = 0

Let f be a differentiable function on [α, β ] and x ∈[α, β ] .Show that, if

f ′(x) = 0 and , f ′′(x) >0 then f must have a local maximum at x.

Let f: [0, 1]→R be a function defined by f(x) = x^m (1-x)^n ,where . m, n∈N

Find the values of m and n such that the Rolle’s Theorem holds for the function

f .

Pio and Sons Inc. Spends P40, 000 to prepare a bid on a construction project. If the contract is awarded, the estimated revenue will be P300, 000. If the contract is not awarded, the company has a penalty of P50, 000. There is 40% chance that the company will be awarded the contract. Find the EV.

Determine the local minimum and local maximum values of the function f defined by

f(x) = 3-5x³ +5x⁴ -x^5

Prove that a strictly decreasing function is always one-one.

Find the following limit.:

Lim x →0 for 1-cos x²/x².sin x²

List down 5 of Random Sampling and 5 examplesof Non Random Sampling. Also, identify if what particular random sampling and non sampling techniques are being applied.

First Column

Random Sampling

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5.

Second Column

Non Random Sampling

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5.