Calculus Answers

Represent (2+ root5 )on the real line.

Using Weierstrass’ M-test, show that the series
Infinity
∑ x/n(n+2)^2
n=1

Converges uniformly in [0,k] where k is any given finite positive number.

Examine the continuity of the function f:[1,4]→R defined by f(x)= [ x]+4/(3x-1)

Find the absolute maximum and minimum values of f(x)=−(x2−x)23 over the interval [-4, 3].

Use the order completeness property to show that the set
S={n/(n+1): n∈N}

has a
supremum and infimum.

Find the exact global maximum and minimum values of the function g(x)=7x−x2−15 if its domain is all real numbers.

The acceleration due to gravity, g, is given by
g=GMr2,
where M is the mass of the Earth, r is the distance from the center of the Earth, and G is the uniform gravitational constant.

(a) Suppose that we increase from our distance from the center of the Earth by a distance Δr=x. Use a linear approximation to find an approximation to the resulting change in g, as a fraction of the original acceleration:
Δg≈ g×

(Your answer will be a function of x and r.)

(b) Is this change positive or negative?
Δg is
(Think about what this tells you about the acceleration due to gravity.)

(c) What is the percentage change in g when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.4 km; assume the radius of the Earth is 6400 km)?
percent change =

Find the upper and lower Riemann integrals of the function f , defined on [a,b]
as follows:

0 when is x irrational
f(x)={
2 when is x irrational

Is f Riemann integrable on [ a,b]?Justify your answer.

Locate the critical points and identify which critical points are stationary points for the function f(x)=5x4−20x .

Find and classify the critical points off(x)=9x4(2−x)5 as local maxima and minima.
Critical points: x