Calculus Answers

Find the directional derivative of :a)f(x,y)=x^2y^3-y^ at (2,1) in the direction θ=π/4 b)f(x,y,z)=squareroot(xyz) at (3,2,6) in the direction of the vector v=< 1,2,2>

Find the equation of the tangent plane to the hyperboloid x^2/a^2 + y^2/b^2 - z^2/c^2 = 1 at (x0,y0,z0)

Find the local maximum and minimum values and saddle point(s) of the function f(x,y)=x^3-12xy+8y^3

Use the method of Lagrange multipliers to find the local extreme values of the function f subject to the constraint:
We introduce the multiplier λand solve the system ,∇f=λ∇g, g=0
i. f(x,y)=xy on the ellipse x^2+4y^2=1
ii.f(x,y)=2x-y+6 on the circle x^2+y^2=

A prison guard tower sits 100 yards from a jail and shines a light across the jail yard at 5 revolutions each minute. This is equivalent to 12 pi radians/minute. How quickly is the spot of light moving across the jail yard when it is 50 yards from the point on the jail yard which is nearest the guard tower?

3x^5 square root x^3+1

♦ Suppose that we don’t have a formula for g but we know that g(2)=-4 and g'(x)= sqt(x^2+5)
-use a linear approximation to estimate g(1.95) and g(2.05)
-solve

♦ Use a Riemann Sum approximation using midpoints for the following definite integrals using
the indicated number of subintervals. Then find the exact area, and compute the error in your
approximation.

-definite integral
a=0, b=3.14 (sin(3x))dx; n=12

Use a Riemann Sum approximation using midpoints for the following definite integrals using
the indicated number of subintervals. Then find the exact area, and compute the error in your
approximation.

- definite integral:
a=0, b=2 (x^3-3x+3)dx;n=8

A spotlight on the ground is shining on a wall 20 m away. If a woman 2 m tall walks from the spotlight toward the building at a speed of 1.2 m/s, how fast is the length of her shadow on the building decreasing when she is 4 m from the building?