Calculus Answers

The acceleration due to gravity, g, is given by
g =
GM
r
2
,
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 cen￾ter of the Earth by a distance ∆r = x. Use a linear approximation
to find an approximation to the resulting change in g, as a frac￾tion of the original acceleration:
∆g ≈ g×
(Your answer will be a function of x and r.)
(b) Is this change positive or negative?
∆g is [?/positive/negative]
(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 4.35 km; assume the
radius of the Earth is 6400 km)?

Let p(x) = x
2
(a ax), where a is constant and a > 0.
Find the local maxima and minima of p.
(Enter your maxima and minima as comma-separated
xvalue,classification pairs. For example, if you found that x =
=2 was a local minimum and x = 3 was a local maximum, you
should enter (-2,min), (3,max). If there were no maximum, you
must drop the parentheses and enter -2,min.)
maxima and minima:
What effect does increasing the value of a have on the x￾position of the maximum(s) you found? (Enter left, none or
right if it moves left, has no effect, or moves right.)
What effect does increasing the value of a have on the x￾position of the minimum(s) you found? (Enter left, none or
right if it moves left, has no effect, or moves right.)
What effect does increasing the value of a have on the y￾coordinate of the maximum(s) you found? (Enter up, none or
down if it moves up, has no effect, or moves down.)

Integrate w.r.t to x using substitution
(A) ∫ x^3√x^4+1 dx from 1 to 0
(B) tan^7x sec^2x dx from pi by 4 to 0
(C) sec^2 (cosx) sinx dx

Derive the reduction formula
∫ (x^2+a^2)^n\2 dx= x (x^2+a^2)^n\2\n+1+na^2\n+1∫ (x^2+a^2)^n\2-1 dx
Use the formula to integrate ∫ (x^2+a^2)5\2

Integrate
(A)∫ (3x+1)√4x^2+12x+5 dx
(B)∫x^2+x+5\(x^2+4)(x+1) dx

Use integration by parts to integrate ∫x secx tanx dx

Find the length of the portion of the parabola x =3t^2,y=6t cut off by the line 3x+y-3=0

Integrate the following functions w.r.t x:
(A) 1\(2x+1)^3\2
(B)sin (2x+3)
(C)cosec (4x)
(D)1\√1-9x^2
(E)1\1+4x^2

The height of an inverted cone as it is filled with sand is always four times the radius. The radius is increasing at the rate of 3 cm/s. Calculate the rate at which the volume is increasing at the instant the volume is 800 cm^3

Differentiate y w.r.t x in the following cases:
(A) y= ln (x ln x)