Calculus Answers

1. (a) Give the definition of continuity of f at a. (b) Give the definition of the derivative of f at a.
2. For the function f(x) =  x 2 − x + 1 1 − x − x 2 if x < 0, if x ≥ 0

(a) Use your definition of continuity in 1(a) and the properties of limits to determine if f is continuous at a = 0. Show your work.

(b) Use your definition of derivative in 1(b) and the properties of limits to determine if f has a derivative at a = 0. Show your work.

For the function f(x) = 2x − 3 x (a) Use the limit definition of the derivative to find f 0 (x). (b) Find the equation of the line tangent to f(x) at x = 3.

 For the function f(x) = x 2 use the Intermediate Value Theorem to prove that there exists a number c such that f(c) = 2. Your are proving the existence of the real number √ 2. Since f(1) < 2 < f(2) you can restrict f(x) to the interval [1, 2]

Let f(x)=−6x2+3x+10,find the slope of parabola f(x)

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

• A 5m ladder leans against a vertical wall. If the top starts sliding downward at the rate of 5.0 ft/sec, find how fast in m/sec the lower end moves when it is 4 m from the wall.

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.