How an Automatic Transmission Changes Gears Smoothly: 

- An automatic transmission uses a torque converter to smoothly transmit power from the engine to the transmission.

- Inside the transmission, there are multiple gear sets and clutches that engage and disengage to change gears.

- The transmission control module (TCM) monitors various sensors to determine the optimal timing and pressure for gear shifts.

- Hydraulic pressure is used to engage and disengage the clutches, allowing for seamless gear changes.

Signs of a failing automatic transmission:

1. Delayed engagement: The transmission takes longer to shift into gear when you shift from park to drive or reverse.

2. Slipping gears: The engine revs up without a corresponding increase in vehicle speed, or there is a sudden loss of power while driving.

3. Rough or jerky gear shifts: Shifting between gears feels harsh, accompanied by vibrations or a jolt.

4. Fluid leaks: Noticeable leaks of reddish-brown fluid under the vehicle.

5. Burning smell: A burnt odor may indicate overheating or excessive friction within the transmission.

6. Warning lights: The "Check Engine" or "Transmission" warning light may illuminate on the dashboard.

Repair and recommendations for a failing automatic transmission:

1. Consult a professional mechanic: Transmission repairs can be complex, so it's best to seek the expertise of a qualified technician.

2. Regular maintenance: Follow the manufacturer's recommended maintenance schedule, including fluid changes and inspections.

3. Check fluid levels and quality: Low or contaminated transmission fluid can cause issues, so ensure it is at the correct level and in good condition.

4. Address issues promptly: If you notice any signs of transmission problems, address them promptly to prevent further damage.

5. Avoid excessive towing or overloading: Heavy loads can put strain on the transmission, leading to premature wear and failure.

DTC codes associated with automatic transmission issues:

1. P0700 - Transmission Control System Malfunction: Indicates a problem with the transmission control module or related components.

2. P0715 - Input/Turbine Speed Sensor Circuit Malfunction: Points to a malfunction in the input/turbine speed sensor, which provides vital data to the TCM.

3. P0720 - Output Speed Sensor Circuit Malfunction: Indicates an issue with the output speed sensor, which monitors the rotational speed of the transmission output shaft.

4. P0730 to P0736 - Incorrect Gear Ratio: These codes indicate a mismatch between the gear ratio commanded by the TCM and the actual gear ratio detected.

5. P0740 to P0744 - Torque Converter Clutch Circuit Malfunction: Points to problems with the torque converter clutch solenoid or related components.