Calculus Answers

If a water wave with length L moves with a velocity 𝑣 across a body of water with depth d, as in figure below,

then

𝑣2 = 𝑔𝐿 tanh 2𝜋𝑑, 2𝜋 𝐿

a) If the water is deep, show that 𝑣 ≈ √𝑔𝐿. 2𝜋

b) If the water is shallow, use the Maclaurin series for tanh to show that 𝑣 ≈ √𝑔𝑑. (Thus, in shallow water the velocity of a wave tends to be independent of the length of the wave.)

Consider the functions

f(x)=5 and g(x)=5x.

f(x)=5 and g(x)=5x.

The value of g(x)−f(x)

g(x)−f(x) is

Find the minimum and maximum values of f(x,y,z)=8x^2-2y subject to x^2+y^2=1

Using wiestrass M test show that the following series converges uniformly Sigma infinity n=1 n^3 x^n x belongs to[-1/3,1/3]

Let f (x)= x²-4 for x<3

7 for x=3

2x+4 for x>3

a.)f(0)=

b.)f(3)=

c.)f(5)=

d )lim f(x) as x approaches to 0=

e.)lim f(x) as x approaches to 3 from left=

f.)lim f(x) as x approaches to 3 from right=

Classify the Critical points of f(x,y)=4+x^3+y^3-3xy

When heating a 15 cm long square metal plate, its size increases by 0.05 cm.

Actions to be performed:

Answer: approximately how much did its area increase?

Determine how much its volume increases if it were a cubic plate.

If the plate is cooled and its side decreases by 0.05, how much did its area decrease?

If it is a cubic plate determined and it is cooled by decreasing its side by 0.05, determine how much its volume decreased?

1) Integrate dx/2√2x^3

1) Find the tangent line to the graph given by x^2(x^2+y^2)=y^2 at the points(√2/2, √2/2)

When a square metal plate 15 cm long is heated, its side increases by 0.05 cm.

If the plate is cooled and its side decreases by 0.05, how much did its area decrease?

If it is a cubic plate and it is cooled by decreasing its side by 0.05, determine how much its volume decreased?