Derivatives
Find dy/dx: y = 3x2 + (2/x) - (5/x2).
What is dy/dx = 6x - (2/x2) + (10/x3)?
∫xdx/(x4 + 1).
What is 1/2 arctan(x2) + C?
Which theorem is this? There exists a point c in the interval (a,b) such that f'(c)=f(b)-f(a)/(b-a) if a function is continuous on the closed interval [a,b] and differentiable on the open interval (a,b)?
What is the Mean Value Theorem?
Water pours into a fish tank at a rate of 3ft3/min. How fast is the water level rising if the base of the tank is a rectangle of 2 x 3 feet?
What is dh/dt=0.5ft/min?
Find dy/dx: y = (2-x)/(3x+1).
What is dy/dx = -7/(3x+1)2?
What is ∫-3-2 5(2x+4)1/3dx?
What is -15(21/3)/4?
What is the first fundamental theorem of calculus?
What is ∫ab f'(x)dx= f(b) - f(a)?
What point on y=2x+5 lies closest to the origin?
What is (-2,1)?
If f(4)=6, f'(4)=2/3, and f'(6)=3/5, find (f-1)'(6).
What is (f-1)'(6) = 3/2?
∫xsinxdx.
What is -xcosx + sinx + C?
Let f(x) = x2 - x. Does there exist some c in (0,1) in which f'(c)=0?
What is yes?
Find the volume of the solid that results when the region enclosed by the curves is revolved about the axis x = 1.
x = √y + 3
x = 3 + y/2
What is 8π?
At what x-value is y=3x-1 tangent to f(x)=x3+1?
What is x=±1?
∫(4-x2)1/2dx.
What is 2arcsin(x/2) + (1/2)x(4-x2)1/2 + C?
Use properties of integrals to find the derivative:
∫√x x3 sinu(√u)du.
What is 3x7/2sin(x3) - (1/2)x-1/4sin√x?
A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5ft/sec. At what speed is the length of the person's shadow changing when the person is 16 ft from the lamppost?
What is 35/13 ft/sec?
Find dy/dx: y = (x + 2)4 * (2x − 5)2 * (5x + 1)3?
What is y(4/(x+2) + 4/(2x-5) + 15/(5x+1))?
What is lim n → ∞ Σi=13 (3/n)*(e3i/n)?
What is ∫03 exdx?
What is a slope field?
What is a differential equation used to visualize a family of antiderivatives?
Find the root of x3 - x2 +3 = 0. Use Newton's method to give x2 an approximation of the root where x1 = -1.
What is -6/5?