An object of mass m moves from x₁ to x₂ during the time interval [t₁, t₂] due to a force F(x) acting in the direction of motion. Let x(t), v(t), and a(t) be the position, velocity, and acceleration at time t. The object’s kinetic energy is KE = 1/2 mv².

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(a) Use the Change of Variables Formula to show that the work performed is equal to S 11 W = ₁²² X1 F(x) dx = dt F(x(t))v(t) dt (b) Use Newton's Second Law, F(x(t)) = ma(t), to show that d (1 ( 1 mv (1)²) = F(x(1))v(1) (c) Use the FTC to prove the Work-Energy Theorem: The change in kinetic energy during the time interval [t₁, 1₂] is equal to the work performed.