Calculus AB review

Limits and Continuity

100

Lim x-> 2 (x² - 4 / x² + 4) is 

0

200

Lim x->∞ (4 - x² / x² - 1) is

-1

300

Lim x->∞ (5x³ + 27 / 20x² + 10x + 9) is

-∞

400

Lim x->-∞ ( 2^-x / 2^x ) is

∞

500

If f(x) = (x² - x / 2x) for x ≠ 0; f(0) = k and if f is continuous at x = 0, then k =

-1/2

Differentiation

100

Y = 3x^2/3 - 4x^1/2 - 2 

Find y’

2x^-1/3 - 2x^-1/2

200

y = (4x + 1)(1 - x)³ 

Find y’

(1 - x)²(1 - 16x)

300

Y = ln(secx + tanx) 

Find y’

Secx

400

Y = (e^x - e^-x)/(e^x + e^-x) 

Find y’

4/(e^x + e^-x)²

500

X³ + y³ - 3xy + y² = 0

Dy/dx = (3x² - 3y)/(3x - 3y² - 2y)

Contextual Applications of Differentiations

100

The object is speeding up if…

velocity and acceleration are both positive or both negative. 

200

S = t³ - 6t² + 12t - 8 

The position of the particle is increasing for 

all t except t = 2

300

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

400

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

500

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

Theorems

100

If g(2) = 12 and g’(x) ≥ 1/2 for 2 ≤ x ≤ 6, what is the smallest value g(6) can be? Why? 

14

200

 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

300

Find the minimum value of the function f(x) - 2/x. 3lnx on the interval (1/2, e)

1.7836

400

Consider the function f(x) = e^-x on the interval (0, ln2). Find the “c” value guaranteed by the MVT. 

X = 0.3267

500

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

Integration

100

∫ 4x⁶ - 2x³ + 7x - 4 dx

4/7x⁷ - 1/2x⁴ + 7/2x² - 4x + c

200

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

300

∫ 1/(3x + 12) dx = 

1/3ln|x + 4| + C

400

F(x) = ∫^x₃ (tan(5t)sec(5t) - 1)dt

Find F’(x)

tan(5x)sec(5x) - 1

500

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