Find the function from derivative and a single value. 1. Find the antiderivative a. Given f'(x) = (5x^4)-(3x^2)+4 and f(-1)=2, find f(x) b. Given f'(t)=t + (1/(t^3)) , t>0 , and f(1)=6 , find f(t) i. Where f(t)=t+1/t^3 find as t + 1(numerator)/(t^3) (denominator) c. Given f'(x) = (x+1)/(sqrt(x)) , and f(1) = 5 , find f(x). i. Where f'(x) = (x+1 (as numerator))/(sqrt(x)denominator)). 1. Sqrt = Square Root Symbol 2. Symbols a. Find f(x) if ((d^2)f)/(dx^2) = 6-18x , f(0)=6 , and f(2)=12 i. Where ((d^2)t) is numerator/(dx^2) is denominator Rectilinear Motion 3. A particle moves in a straight line and has acceleration given by a(t) = sin(t). Find the equation for the position of the particle s(t), given that x(pie/2) = 1 and s(0) = 0 (where v(t) is velocity at time t) a. Where v(pie(numerator)/2(denominator) i. Pie is reference to Pi Symbol. 4. A particle moves in a straight lune and has acceleration given by a(t) = sin(t) +cos(t). Find the equation for the position of the particle s(t) , given that v(0) = 4 and s(0) = 3 (where v(t) is velocity at time t ). Riemann Sums 5. Integration a. Approximate the are under the curve y = x^2 +3 on [0,2] with a Riemann sum using 4 sub-intervals and midpoints. b. How wude should each sub-interval be for approximating the are under the curve for y=2*(sqrt(2*x)) on [2,4] using 4 rectangles? i. Where (2*x) is within the mouth of the Sqrt. 1. Sqrt = Square Root Symbol c. Approximate the are under the curve for 2 * (sqrt(2*x)) on [2,4] with a Riemann sum using 4 sub-intervals and right enpoints. i. Where (2*x) is within the mouth of the sqrt 1. Sqrt = Square Root Symbol d. Approximate the area under the function f(x) , based ont eh given data, using left endpoints. 6. A car starts moving at time t=0 and goes faster and faster. Its velocity is shown in the following table: Estimate how far the car travels during the 12 seconds. Definite integrals via geometry 7. Integration a. The sketh below represents the signed are of definite integral . Evaluare the integrak using an appropriate formula from geometry. b. The sketch below represents the signed area of the definite integral . Evaluate the integral using and appropriate formula from geometry. c. The sketch below represents the signed area of the definite integral . Evaluate the integral using an appropriate formula from geometry. 8. Use geometry to compute the following definite integral: 9. Integration a. Given the following: Find the value, if possible of: . 10. Given the following: . Find the value if possible of: . 11. Given , evaluate