Limits and Continuity
Lim x-> 2 (x2 - 4 / x2 + 4) is
0
Y = 3x2/3 - 4x1/2 - 2
Find y’
2x-1/3 - 2x-1/2
The object is speeding up if…
velocity and acceleration are both positive or both negative.
If g(2) = 12 and g’(x) ≥ 1/2 for 2 ≤ x ≤ 6, what is the smallest value g(6) can be? Why?
14
∫ 4x6 - 2x3 + 7x - 4 dx
4/7x7 - 1/2x4 + 7/2x2 - 4x + c
Lim x->∞ (4 - x2 / x2 - 1) is
-1
y = (4x + 1)(1 - x)3
Find y’
(1 - x)2(1 - 16x)
S = t3 - 6t2 + 12t - 8
The position of the particle is increasing for
all t except t = 2
f’(x) = (-3 - x2)/x2 for x < 0.. The the x = c value(s) for f(x) on the interval (-3, -1) guaranteed by the MVT given that f(-1) = -2 and f(-3) = 2
x = -√3
For a certain continuous function f, the right Riemann sum approximation of ∫20 f(x)dữ with n sự intervals of equal length is (2(n+1)(3n+2)/n2) for all in. What is the value of ∫20 f(x)dx
6
Lim x->∞ (5x3 + 27 / 20x2 + 10x + 9) is
-∞
Y = ln(secx + tanx)
Find y’
Secx
The side s(t) of a square is decreasing at a rate of 2 kilometers per hour. At a certain instant t0 , the side is 9 kilometers. What is the rate of change of the area A(t) of the square at that instant?
A(t) = (s(t))2
Find the minimum value of the function f(x) - 2/x. 3lnx on the interval (1/2, e)
1.7836
∫ 1/(3x + 12) dx =
1/3ln|x + 4| + C
Lim x->-∞ ( 2-x / 2x ) is
∞
Y = (ex - e-x)/(ex + e-x)
Find y’
4/(ex + e-x)2
The side of a cube is decreasing at a rate of 9 millimeters per minute. At a certain instant, the side is 19 millimeters.What is the rate of change of the volume of the cube at that instant (in cubic millimeters per minute)?
-9747 ml3/min
Consider the function f(x) = e-x on the interval (0, ln2). Find the “c” value guaranteed by the MVT.
X = 0.3267
F(x) = ∫x3 (tan(5t)sec(5t) - 1)dt
Find F’(x)
tan(5x)sec(5x) - 1
If f(x) = (x2 - x / 2x) for x ≠ 0; f(0) = k and if f is continuous at x = 0, then k =
-1/2
X3 + y3 - 3xy + y2 = 0
Dy/dx = (3x2 - 3y)/(3x - 3y2 - 2y)
A 10-ft ladder leans against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house at 2 ft/s. At what rate is the angle between the ladder and the ground changing when the base is 8 ft from the house?
-1/3 rad/s
Consider the function h(x) = 3x/π + cosx. Since h(x) is continuous on (0, π/2) and differential equations on (0, π/2), the MTV applies. Find x = c, 0<c<π/2, that satisfies the conditions of the MVT.
c = 0.690
The number of bacteria in a container increases at the rate of R(t) bacteria per hour. If there are 1000 bacteria at time t = 0, which of the following expressions gives the number of bacteria in the container at time t = 3 hours?
1000 + ∫30 R(t)dt