Calculus Answers

Calculate a unit vector normal to the surface
x3 + y3 + 3xyz = 3 at the point (1, 2, - 1).

A rigid body is rotating with an angular speed of
3.0 rad s-1 about an axis OL = 21 -2j + k,
where 0 is the origin. Determine the velocity of
the body at the point P(4, 1, 2).

What is the integral of 3^x e^x?

Find lcm of following:
1) π and 2π
2).3π/4 and 5π/4

b. Sketch the graph of a single function f that satisfies all of the following conditions:
i) lim f(x)=-2 as x approaches neg infinity
ii) lim f(x) = 0 as x approaches infinity
iii) lim f(x) = infinity as x approaches -3 from the right
iv) lim f(x)= neg infinity as x approaches -3 from the left
v) lim f(X) = -2 as x approaches 3 from the right
vi) f is continuous from the right at 3

1) Find the result of the following integrals.
a) ∫▒〖(2+y)^2 dy〗
b) ∫▒(x²+x+1)/√x dx

2) Find the results of the following integrals by using Substitution rules.
a) ∫_0^1▒〖xe〗^(-x²) dx
b) ∫_0^1▒〖(e^z+1)/(e^z+z) dz〗
c) ∫▒〖dx/(ax+b)(a≠0)〗

By using the definition, f^' (a)= lim┬(n→∞)⁡〖((f(a+h)-f(a))/h)^ 〗, find the derivatives of f=((2a+1))/((a+3))

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = quantity x minus eight divided by quantity x plus seven and g(x) = quantity negative seven x minus eight divided by quantity x minus one.

1) At 2.00 pm, a car’s speedometer reads 30 mi/h. At 2.10 pm it reads 50mi/h. Show that at some time between 2.00 pm and 2.10 pm the acceleration is exactly 120 mi/h²

2) Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f(t)=g(t)-h(t) ,where g and h are the position functions of the two runners.]

A 6 foot boy throws the javelin with an initial speed of 87 feet per second at an angle of 38° with the horizontal. Mark threw approximately 180’ at the Pasco Invite. Does the boy described above beat Mark?