Calculus Answers

 Let 𝑓(𝑡) = 𝑒 −(𝑡−1) 2 describes the position of a particle at time 𝑡 ≥ 0. a) What is the initial displacement? b) Find the critical points of 𝑓(𝑡). c) Find the intervals of positive and negative velocities of the particle. d) Find the time where particle changes from acceleration to deceleration and vice versa. e) Find the maximum displacement of the particle. f) Sketch the graph of 𝑓(𝑡). [

𝑓(𝑥) = 𝑒 −𝑥 1+𝑒−𝑥 . i) Determine whether 𝑓(𝑥) is a one-to-one function. 

Let f : R²→R,f(x,y) =2xy/x²+y² When (x,y) ≠(0,0) and f(x,y) =0, otherwise. How to Check whether or not f has a directional derivative at (0, 0) in the direction θ=π/4? Deduce that the function f is not differentiable at the point (0, 0)?

Suppose that a certain market supplies x thousands of crates of orange daily when p dollars is the price per crate, and the supply equation is px-20p-3x+105=0. If the daily supply decreases at the rate of 250 crates per day, at what rate is the price changing when the daily supply is 5000 crates? 

The demand equation for a particular kind of shirt is 2px+65p-4950=0, where x hundreds of shirts are demanded per week when dollars is the price of a shirt. Suppose that the shirt is selling this week at $30, and the price increases at the rate of $0.20 per week. Find the rate of change in demand. 

A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the

other is bent into an equilateral triangle. How should the wire be cut so that the total area

enclosed is:

(a) maximum? (b) minimum?

find workdone integral of F



d

~

r

where

~

F

(

x

,

y

,

z

)

=

e

2

x

~

i

+

z

(

y

+

1

)

~

j

+

z

3

~

k

along the curve

C

~

r

(

t

)

=

t

3

~

i

+

(

1

−

3

t

)

~

j

+

e

t

~

k

for 0

≤

t

≤

2

Calculate the approximate value of √26 to four decimal places using Taylor series method.

4.Expand log (cosx) about the point.