Calculus Answers

If (𝑥, 𝑦, 𝑧) =𝑋^ 2 𝑌 − 2 𝑌^ 2𝑍 ^3, find ∇𝑓 at the point (1, −1,2)

the sum of two numbers is 4. find the minimum value of the sum of their cubes.

A. Tell whether each given function has a solution on the indicated closed interval. Prove using the Intermediate value theorem.

1. f(x)=3x²+2x²;[-1,1]

2. f(x)= 2-x²/x²;[-3,-1]

B. Sketch the graph of the following functions and then find the absolute extreme values of each of the given interval.

1.f(x)=√x²-25;[5,10]

2. f(x)=-1/x²;[0.5,2]

C. A restaurant's profit function (in hundreds) for hamburgers is given by the function P such that P(x)=1.22x-x²/30,000-4,000,where 0≤x≤20,000.

1. How many hamburgers does the restaurant need to sell to yield the maximum profit?

2. What is the maximum profit from the sale of hamburger ?

B. Sketch the graph of the following functions and then find the absolute extreme values of each of the given interval.

1.f(x)=3-2/5x;[-4,0]

2.f(x)=x-3/4+x;[-3,1]

C. Let f(x)=2x+1. Determine if the Intermediate value theorem applies to f on the closed interval [-3,4]for k=1.

using ϵ−δ definition, show that

lim _x→2 (1/2 x² - x + 1)

Using the definition of limit at infinity or infinite limits, prove that

a) lim_{𝑥 → 3} 1/ (𝑥 − 3) ² = ∞ 

 Using Intermediate Value Theorem, show that

x2 = √𝑥 + 1 has a root in (1,2)

Find the Derivative of the following function

1. y = x² cos x - 2xsinx - 2cosx
2. x(cos2x + 3y) = ysinx

Find the derivative of a function using Limit definition of derivative.

y = 4 √x

Given functions 𝑓(𝑥) = 6 + √𝑥 − 4 and 𝑔(𝑥) =

10

𝑥−5

. 

Determine the domain and range of 𝑓(𝑥) and 𝑔(𝑥)