Calculus Answers

TOPIC: General Application of Derivatives. Draw the necessary figure and indicate the dimension given.

1. The perimeter of an isosceles triangle is 28 cm. What is the maximum area possible?

TOPIC: General Application of Derivatives. Draw the necessary figure and indicate the dimension given

1. A closed cylindrical can is to have a volume of 130 cm³ and a minimum total surface area. Find the base radius R and the altitude H of the can.

Let ℒ{𝑓(𝑡)} = 𝐹(𝑠). Show that 𝑓(𝑡) = − 1 𝑡 ℒ −1 {𝐹 ′ (𝑠)} Thus, if we know how to invert 𝐹 ′ (𝑠) then we know how to invert 𝐹(𝑠). Use this information to find the Laplace inverse transform of (i) arctan ( 𝑎 𝑠 ), (ii) ln ( 𝑠+𝑎 𝑠−𝑎 ). 

With the use of Laplace transform, solve the following I.V.Ps (i) 𝑦 ′′ + 2𝑦 ′ + 𝑦 = 𝑒 −𝑡 , 𝑦(0) = 1, 𝑦 ′ (0) = 3, (ii) 𝑦 ′′ + 𝑦 = 𝑡 2 sin 𝑡, 𝑦(0) = 𝑦 ′ (0) = 0’ (iii) 𝑦 ′′ + 3𝑦 ′ + 7𝑦 = cos 𝑡, 𝑦(0) = 0, 𝑦 ′ (0) = 2, (iv) 𝑦 ′′ + 𝑦 ′ + 𝑦 = 𝑡 3 , 𝑦(0) = 2, 𝑦 ′ (0) = 0.

Find the inverse Laplace transform of the following (i) 𝑠 (𝑠+𝑎) 2+𝑏2, (ii) 𝑠 2−5 𝑠 3+4𝑠 2+3𝑠 ’ (iii) 3𝑠 (𝑠+1) 4, (iv) 1 (𝑠 2+1) 2. 

Find the center of mass of a triangular sheet whose vertices are (8, 0), (2, 6) and (2, 0), whose density function is

p (x, y) = x + y + 1

Integral sech √x tanh √x/√x dx

f(x)=(3x^(3)+5)*e^(2x^(2))