Know Your Limits
f(x) = (4x3 + 7x)/(8x5 + 8x2 + 1)
Find the limit as x approaches ∞.
The limit of f(x) as x approaches ∞ is 0
What is the derivative of F(x) if
F(x) = x3 + 12x + 3x-1?
F'(x) = 3x2 + 12 - 3x-2
Find the integral of f'(x) if f'(x) = 4x3 + 4x + 3.
f(x) = x4 + 2x2 + 3x + C
What is the derivative of g(t) = 3x3 + 9x2 + 8x at t=2 minus the derivative of b(t) = -2x6 + 6x4 + 76x2 at t=1? Round the answer to the nearest whole number.
g'(t) - b'(t) ≈ -84
What is the equation of the slope of tangent line on f(x) = 3x2 - 9x + 3 when x = 5?
NOT Calculator Active
y - 33 = 21(x - 5)
If f(x) = (8x4 + 2x2 + 16)/(64x4 - 7x2 - 72), what is the limit? Simplify.
The limit is 1/8.
What is the derivative of f(x) if f(x) = 4/x3 + 6x1/2 + 8x?
f’(x) = -12/x2 + -3/x1/2 + 8?
Find the integral of f'(x) if f'(x) = 3sin(x/2).
f(x) = -6cos(x/2) + C
Region A is the area enclosed by the function f(x), f(x) = x3 - 2, y = -1 and x = 0. Find the volume of the solid generated when the region A is revolved about the line y = -1. Round to the nearest tenth.
V ≈ 2.0 cubic units
The velocity of a particle at time t is given by v(t) = 2/t2 + 5. If the position of the particle is x = 4 when t = 3, what is the position of the particle when t = 6? Round to the nearest thousandth.
Calculator Active
x(6) ≈ 19.333
3x2 + 3x - 7, x > 2
f(x) = { 10, x = 2
6x - 1, x < 2
Is f(x) continuous at x = 2?
No, f(x) is NOT continuous at x = 2 because the limit is 11 but f(2) = 10
What is the derivative of f(x) if
f(x) = (7x + 1)(4x2 + 3x)?
f'(x) = (7x + 1)(8x + 3) + (4x2 + 3x)(7)
What is the integral of
y' = 45x3 - x-1 + 2x + 89.76?
y = 45x4/4 - ln|x| + x2 + 89.76x + C
Let the region R be the base of a solid, cross sections cut perpendicular to the x-axis are isosceles right triangles. The right R is bounded by the function f(x) x3 - 1 , y = 7 and the y axis. Write but do not evaluate an integral expression for the volume of the solid.
V = 1/2∫(0 to 2) [(7 - x3 + 1)2] dx
What is the limit of f(x) at x -> 2 for
(4x2 - 8x) / (x3 - 2x2)?
NOT Calculator Active
lim f(x) at x -> 2 = 2
What value of k will make the function
f(x) = {12x + 6, x>4
kx - 3, x<4
continuous?
k = 57/4
If g(x) and h(x) are inverses find h'(x) when g(x) = tan(x).
h'(x) = 1/sec2(x)
or
h'(x) = cos2(x)
What is the integral of f'(x) = 5x/3 (5x2 + 3)3?
f(x) = 1/24 (5x2 + 3)4 + C
Sand is deposited into a pile with a circular base. The volume V of the pile is given by V = r3/3, where r is the radius of the base, in feet. The circumference of the base is increasing at a constant rate of 5π feet per hour. When the circumference of the base is 8π feet, what is the rate of change of the volume of the pile, in cubic feet per hour?
dV/dt = 40 cubic feet/hour
If a(t) = 3t2 + 8t + 30, what is the position at t = 2?
NOT Calculator Active
896/12 or 224/3(simplified)
Find the limit
f(x) = (4x - x2)/(2 - x1/2) as x approaches 4.
16
What is the derivative of g'(x) if
g(x) = e(12x - 2)(7x + 1)?
g'(x) =
[7(12x - 2) + 12(7x + 1)]e(12x - 2)(7x + 1)
What is the integral of g'(x) = x2 + 3x - 1 if the upper limit is 3 and the lower limit is 1?
56/3
Region A is the area enclosed by the function f(x) = f(x) = x1/3 and g(x) = 0.5x. Find the volume of the solid generated when the region A is revolved about the line y = -2. Round to the nearest tenth.
V ≈ 17.3 cubic units
Find dy/dx of 5x2y + 3y2 = 5 - 3x + 7y when y = -1 and x = 2?
Calculator Active
dy/dx = (10xy + 3)/(7 - 5x2 - 6y)
OR
dy/dx = (-3 - 10xy)/(5x2 + 6y - 7)
= 17/7