Calculus Answers

(3) (i) A broken pipe at an oil rig off the east coast of Trinidad produces a circular oil slick that is S meters thick at a distance x meters from the break. It difficult to measure the thickness of the slick directly at the source owing to excess turbulence, but for x >0 they know that s(x) = x^2/2 + 5/3x/x^3 + x^2 + 2x If the oil slick is assumed to be continuously distributed, how thick is expected to be at the source? (ii) If f(x) = {4x + 7 1< x < 2, 4x^2 - 1 2< x < 4 , determine whether the function f (x) is continuous throughout its domain? 

(2) For the function f(x) = x^2/3 - x^2 - 24x + 19 (i) Determine the turning points and the values of the function at each turning point. (ii) Determine the point of inflection, any exist. (iii) What is the range of value over which the function is strictly increasing (iv) What is the range of values over which the function is concave upwards. (3) A production manager determines that t months after production on a new product begins, the number of units produced will be N thousand, where N(t) = 17t^2 + 13t/(2r + 1)^2 . (i) What would be the level of production 5 months into operations. (ii) What would be the level of production be in the long run.

3) The equation for the instantaneous voltage across a discharging capacitor is given by the equation

𝑦 = 12𝑒 ^-x/2

a) Differentiate the function to find an equation for the gradient and calculate two gradients at x = 2 and x = 4.

4)

𝑦 = 10 𝑙𝑜𝑔3x

a)Differentiate the function to find an equation for the gradient and calculate two gradients at x = 2 and x = 8.

5) The displacement, 𝑦(m), of a body in damped oscillation is

𝒚 = 𝟐𝒆 −𝒕 𝐬𝐢𝐧 𝟑𝒕

a) Use the Product Rule to find an equation for the velocity of the object if 𝑣 = 𝑑𝑦 𝑑𝑡

6) You are given the following function which describes the readout taken from an oscilloscope

𝑦 = 3cos (2𝑥)

Your tasks are to:

a) Create a graph showing 2 full periods of the function

b) Take two gradients from the graph

c) Differentiate the function and prove your answers from part b using calculus

d) Compare your answers 

y = x (x-3)⁴

The functions f and g are defined by f(x) =1/(1-3x) and g(x) =log_1/3(3x-2)-log₃(x) respectively

1. Write down the sets D_{f }(ehe domain of f) and D_g (the domain of g)

2. Solve the inequality f(x) > 2 for x"\\isin" D_f

3. Solve the inequality f(x) ≥ 2 for x"\\isin" D_g

Hint: Use the change of base formula

Fatima is having a party and needs the following:

· 27 chocolates

· 25 Packets of chips

When answering the following questions, always write the ratios in simplest form.

3.1 What is the ratio of the number of chocolates Fatima buys to the number of chocolates she gets for free?( 1 )

3.2 What is the ratio of the number of packets of chips Fatima buys to the number of packets she gets for free?( 2 )

3.3 How many chocolates should Fatima buy so that she has 27 chocolates in the end?( 1 )

3.4 How many packets of chips should Fatima buy so that she ends up with 25 packets?( 1 )

3.5 What is the total price Fatima pays for all her goods?( 3 )

Show that the maximum value of a 2 b 2 c 2 on a sphere of radius r centered at the origin of a Cartesian abc-coordinate system in (r 2/3)3 .

The rule of a function f is given. Write an algebraic formula for f(x).

a. Square the input, subtract 16, and take the square root of the result.

b. Cube the input, multiply by 3, add 5, and divide the result by the input. 

Prove that if f is continuous, then:

∫ (x>0) f(u)(x-u).du= ∫ (x>0) ( ∫ (u>0) f(t) dt) du.