# Differentiation and Limits

Not what you're looking for?

See the attached file.

1. Differentiate

a. Y = 3x + PI^3

b. Y = 1 / (x-3)^3

c. y = (x^4 - x)^3 (3x + 2)^4

d. Y = (1 + x - x^3)^4

2. Compute the following limits.

a. lim(x??)?[(x-2)/(x^2+2)]

b. lim(x??)?[(3x^5- 6x^4+ 2x-6)/(7x^5- 2x^2+ 10,000)]

3. Use limits to compute f"(3) where f (x) = x^2 - 2x +3.

4. a. What is the average rate of change of f(x) given f(x) = -6/x from [1,2] and [1,4].

b. What is the instantaneous rate of change of f(x) when x = 1.

5. Write the equation of the tangent line to the curve y = x^3 - 2x^2 +5 at x = 2.

##### Purchase this Solution

##### Solution Summary

Differentiation and limits are discussed in the solution.

##### Solution Preview

Differentiate

y = 3x + π^3

dy/dx=3(1)+0=3

y = 1 / (x-3)^3

y=〖(x-3)〗^(-3)

dy/dx=-3(x-3)^(-4) (1)=(-3)/〖(x-3)〗^4

y = (x^4 - x)^3 (3x + 2)^4

dy/dx=(x^4-x)^3 (4) (3x+2)^3 (3)+(3x+2)^4 (3) (x^4-x)^2 (4x^3-1)

dy/dx=12(x^4-x)^3 (3x+2)^3+3(3x+2)^4 (x^4-x)^2 (4x^3-1)

Note that this can be simplified further

dy/dx=3(x^4-x)^2 (3x+2)^3 [4(x^4-x)+(3x+2)(4x^3-1)]

dy/dx=3(x^4-x)^2 (3x+2)^3 (16x^4+8x^3-7x-2)

y = (1 + x x^3)^4

dy/dx=〖4(1+x-x^3 )〗^3 (1-3x^2)

Compute the following ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts