Calculus Answers

Evaluate:

a) ∫∫D (e^(y^2) + 1) dA where D is the triangle with vertices (0,0), (-2,4) and (8,4).

b) ∫∫D x^(5)sin(y^4) dA where D is the region in the 2nd quadrant bounded by y =3x^2, y = 12 and the y-axis.

The velocity of a particle moving on the x-axis is given by v(t) = t^(3) − 6t^(2) for the time interval 0 ≤ t ≤ 10.

a) When is the particle farthest to the left?

b) When is the velocity of the particle increasing the fastest?

Evaluate ∫∫∫E 6z^2dV where E is the region below 4x + 2y + 2z = 10 in the first octant.

Find favg for the functions given on the interval and determine the value of c in the given

interval for which f(c) = favg.

a) f(x) = 9 − 2e^(4x+1) on [2,6]

b) 8 − cos (x/4) on [0 4π]

Find the distinct interval of length 1 containing a root or solutin of f(x) = x³ - 3x + 5 using IVT

The acceleration of an object moving in a strange way has

been modelled as:

a = x ∙ e^x

Use integration by parts to find an equation to model the

velocity, v, given that v = ∫ x ∙ e^x dx

If In=int_0 to ♾️ {(e^-x)(sin^n) (x) dx } , prove that (1+n^2)In=n(n-1)In-2 for n≥2 .

Find the equations of the tangents to the graph of y=x+1/x that are parallel to y+2x=0

Find the equation of the tangent to at the point where , in f(x)=x^2+4x-5

standard form.

Find the equation of the tangent line to a curve y=-x²-1 that is parallel to the line 2x+y=6.