Calculus Answers

ACTIVITY IN BASIC CALCULUS

QUOTIENT RULE

  I. Find the derivative of the following functions below using the quotient rule. Show your complete solution.

1. "\\frac{x^2-7x}{2x^3+5x^2}"
2. "\\frac{3x^2+7x-14}{x^3-4x^2}"
3. "\\frac{2x^3+8x^2}{x^2+6x^2-10}"

II. Create your own given problem involving quotient rule and solve. Show your complete solution. Do not copy the given example below.

1. Example must have two different terms in numerator, and three different terms in denominator

eg. (do not copy)

y= "\\frac{8x^2-3x}{x^2+6x^2-10}"

2. Example must have three different terms in numerator, and three different terms in denominator

eg. (do not copy)

y= "\\frac{x^2+8x^2-3x}{2x^3+6x^2-10}"

Suppose z is a function of x and y, and tan√y²+x² = z^xe^6y. Solve for ∂z/∂x and ∂z/∂y.

Determine all the relative minimum and maximum values, and saddle points of the function g defined by g(x,y) = x³ - 3x +3xy²

Show that the function f defined by

f(x,y) = {1, (x,y) = (0,0)

{(x² + y)/(x + y), (x,y) ≠ (0,0)

is not continuous by using two path test at (0,0).

What are the sum rule, the product rule, and the quotient rule. 2 examples

ACTIVITY IN BASIC CALCULUS

QUOTIENT RULE

  I. Find the derivative of the following functions below using the quotient rule. Show your complete solution.

1. "\\frac{x^2-7x}{2x^3+5x^2}"
2. "\\frac{3x^2+7x-14}{x^3-4x^2}"
3. "\\frac{2x^3+8x^2}{x^2+6x^2-10}"

II. Create your own given problem involving quotient rule and solve. Show your complete solution. Do not copy the given example below.

1. Example must have two different terms in numerator, and three different terms in denominator

eg. (do not copy)

y= "\\frac{8x^2-3x}{x^2+6x^2-10}"

2. Example must have three different terms in numerator, and three different terms in denominator

eg. (do not copy)

y= "\\frac{x^2+8x^2-3x}{2x^3+6x^2-10}"

Determine the dimensions of the right circular cylinder of

greatest volume that can be inscribed in a right circular cone of

radius 6 cm and height 9 cm.

Find the first partial derivative of

f(x,y) = "(4x-y)\/(4x+y)" at the point (4,2) for both x and y.

5. Decompose

(i)

x2 +x +1

(x +3)(x2 −x +1)

(ii)

x4 −x3 −2x2 +4x +1

x (x −1)2