A blackjack scanner near the left entrance flashes this graph of f(x). You must evaluate the limit the scanner is using before the gate will open.

This is the graph of f(x).
Find: lim as x → 0 of f(1 − √(x²))

A progressive slot machine grows according to a limit expression shown on its maintenance screen. Compute the long-run jackpot value to release the next chip tray.

Find: lim as x → ∞ of (1 + 5/x)^x

A poker analytics monitor tracks three changing casino functions f, g, and h. Use the data table to compute the derivative that controls the table's automatic shuffle lock.

Given the table:
x | f(x) | f'(x) | g(x) | g'(x) | h(x) | h'(x)
1 | 4    | -2    | 3    | 5     | 2    | -1
2 | -1   | 6     | 1    | -3    | 5    | 2
3 | 2    | 1     | -2   | 4     | 6    | -5

p(x) = x² f(g(x)) / (h(x) + g(x))
Find p'(2).

A hidden card pattern on the felt follows an implicit curve. The dealer's scanner is calibrated at one highlighted point, and you need the slope there to expose the next key.

Find dy/dx if (x + y²)² = x at the point (0.25, 0.5).

A ceiling camera sweeps across the roulette floor according to an inverse trig model. Find its instantaneous turning rate so your team can slip past the blind spot.

Find dy/dt if y = sin⁻¹(1 − 3t²) at t = 0.5.

A laser control panel in the vault hallway combines three accumulated signal values. Use the definite integrals below to compute the panel's final reading.

Let ∫ from 2 to 5 f(x) dx = -3,
∫ from 2 to 5 g(x) dx = 7,
and ∫ from 1 to 3 h(x) dx = 2.

Determine:
∫ from 2 to 5 [-2f(x) + 4g(x)] dx − ∫ from 1 to 3 h(x) dx

The vault keypad updates its code using a continuously changing rate. Evaluate the definite integral to reveal the next numeric release value.

Find:
∫ from 0 to 4 [6x/(x² + 6)] dx

A rotating vault wheel is driven by a trig-based motion pattern. Find the antiderivative that models the wheel's total angular change.

Find:
∫ 12x² cos(2x³ − 5) dx

The casino protects the main vault using a laser sensor that is placed 30 feet from the base of the vault wall. A thief is climbing vertically up the vault wall at a rate of 12 ft/sec. The laser sensor on the ground rotates upwards to track the thief. When the thief is 16 feet above the ground, how fast is the angle of elevation of the laser sensor changing? Round your answer to three decimal places.

The casino, which is open 24/7, uses large amounts of electricity to operate. The table below gives the rate at which electricity is consumed at different time intervals.

Elapsed time since 12 A.M. (hrs): 0, 3, 7, 9, 14, 18, 21, 23, 24
Rate (kW): 850, 750, 700, 650, 650, 700, 900, 950, 850

Use the Trapezoidal Rule to approximate the total amount of electricity the casino consumes during this one-day period in kilowatt hours.

A security guard is sprinting down a hallway to catch an intruder, which is you, in the vault. What is the total distance traveled by the guard down the hallway from 0 to 5 seconds if his velocity in feet per second is modeled by the function v(t) = 3t² − 18t + 24?

A big trophy for a poker tournament is being designed, and the base of the trophy is given by the region R in the first quadrant bounded by the graphs of f(x) = √x and g(x) = x/2. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the trophy.