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Find the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \) and evaluate it at \( x = 2 \).
The derivative is \( f'(x) = 9x^2 - 10x + 2 \).
Evaluate at \( x = 2 \): \( f'(2) = 9(2)^2 - 10(2) + 2 = 36 - 20 + 2 = 18 \).
The integral \( \int (4x^3 - 2x + 1) \, dx \) is:
So, the integral is \( x^4 - x^2 + x + C \).
A projectile is launched with an initial velocity of 50 m/s at an angle of 30 degrees. Find the maximum height reached.
The vertical component of velocity is \( 50 \sin(30^\circ) = 25 \) m/s.
Maximum height \( h = rac{(25)^2}{2 imes 9.8} = rac{625}{19.6} pprox 31.887 \) meters.
#### 31.887Certainly! Here are ten advanced high school-level mathematics questions with step-by-step solutions:
A cylindrical tank with a radius of 5 meters is filled with water to a height of 10 meters. If the water is drained until the height is reduced by 2 meters, what is the volume of water removed?