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Find all the antiderivatives of the following function. Check your work by taking the derivative. Gm=5 For the function f, given below, find the antiderivative F that satisfies F(1) = 0. 4 3 f(x)=x -4x_ -2 E) The antiderivative that satisfies the given condition is F(x) = D. Find the solution of the following initial value problem. 1 g/(x)=3x[x2]; 9(1)=2 The solution is g(x) = D. Find the solution of the following initial value problem. v'(x)=8x3 8X ;v(8)=53,x>0 The solution of the initial value problem is v(x) = D. Given the following velocity function of an object moving along a line, find the position function with the given initial position. v(t) =4t+ 5; 5(0) = 0 E) The position function is s(t) = D. a. Determine whether the Mean Value Theorem applies to the function f(X) = In 13x on the given interval [1 ,e]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. Given the following acceleration function of an object moving along a line, find the position function with the given initial velocity and position. a(t) = - 40; v(0) = 22, s(0) = 0 . . . s(t) = (Type an expression using t as the variable.)Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v'(t) = g, where g = 9.8 m/sZ. a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. 0. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A softball is popped up vertically (from the ground) with a velocity of 34 m/s. Find the function F that satisfies the following differential equation and initial conditions. F"(X)= 1, F'(0)= 13, F(O)=11 The function is F(X) = D. Find the function F that satisfies the following differential equations and initial conditions. F"'(X)=18X,F"(O)= 0, F'(0) = 3, F(0) = 9