Find the function whose tangent has slope 𝑥 3 − 2 𝑥 2 + 16 for each of x and whose graph passes through the point (2,5).

The function whose tangent has a given slope is the integral of that slope. The slope given is x^3 - 2x^2 + 16.

Step 1: Integrate the slope function to find the original function.

∫(x^3 - 2x^2 + 16) dx = (1/4)x^4 - (2/3)x^3 + 16x + C

This is the general form of the function. The constant C can be found by using the point that the function passes through.

Step 2: Substitute the point (2,5) into the function and solve for C.

5 = (1/4)(2)^4 - (2/3)(2)^3 + 16(2) + C 5 = 4 - 16/3 + 32 + C 5 = 36/3 - 16/3 + 96/3 + C 5 = 116/3 + C

Step 3: Solve for C.

C = 5 - 116/3 = 15/3 - 116/3 = -101/3

So, the function is f(x) = (1/4)x^4 - (2/3)x^3 + 16x - 101/3.