3.4 Applications of the Derivative - Penn Math

1 2/19/20121 math 103 Applications of the Applications of the DerivativePhysicsposition, velocity, and acceleration()s t()s t ()s t ()321245s tttt= +)Find the velocity at time .at)Find the initial velocity and the velocity after ) When is the particle at rest?c) When is the particle moving forward?d()232445s ttt = +()0s ()2s ()450s =( )( )( )223 224 245s = +12 48 45= +9=()0s t =()232445 0s ttt = +=()238150tt +=()()3350tt =3,5tt ==()0s t >()s t 035+ + + + + + + + +movingforwardmovingforward() ()Moving forw0, 3ard5, ()Moving back wards 3, 5 math 103 Applications of the Derivative )Find the acceleration at time .et)When is the acceleration 0?

2 G)Find the acceleration after 2 ()321245s tttt= +()232445s ttt = +)When is the particle speeding up, when is it slowing down?h()624s tt = ()212 24s = 12= ()624 0s tt = =4t =velocity and accelerationhave same signvelocity and accelerationhave different signs()s t 035()s t 4+ + + + + + + + + + + + + + + + + ++ + + ()s t 40+ + + + + + + + +slowing downslowing downspeeding upspeeding up()0, 3()4, 5()3, 4()5, speeding upslowing down2/19/20122 math 103 Applications of the Derivative )Draw a picture that describes the motioni)Find the total distance traveled by the particle during the first 7 ()321245s tttt= +( )( )( )233312 345 3s= +27 12 9 45 3= + 27 108 135= +108 162= +54=( )( )( )

3 235512 545 5s= +125 12 25 45 5= + 125 300 225= +50=050540t=3t=5t=[]0, 3 :distance traveled54=[]3, 5 :distance traveled4=[]5, 7 :( )( )( )237712 745 7s= +343 588 315= +658 588= 70=707t=distance traveled20=78 math 103 Applications of the DerivativeEconomics()the cost of producing units of a productC xx=()measures the rate of change of the cost function, it is called the .C x =marginal cost() xxxx= ++() xxx = +()()()2983000300030002001, 000, 000100C = +9 9, 000, 000 8 30002001, 000, 000100 = +()300081 240 200C = +41=()4130001 CCx == When producing 3000 units, if you decide to produce 1 more unit, cost will increase by $ cost of the 3001st unit is approximately $ 41.

4 ()Find 3000 and interpret ()()Actual cost of the 3001st unit30013000$ =2/19/20123 math 103 Applications of the Derivative ()4 30003000300100R = +()3000120 300R = +180=()When producing 3000 units, if you decide to produce and sell 1 more unit, revenue will increase by $ revenue from selling the 3001st unit is approximately $ 180.() xxx= +() xx = +()Find 3000 and interpret ()()Actual revenue from selling the 3001st unit30013000$ =()the revenue gained by selling units of a productR xx=()measures the rate of change of the revenue function, it is called the .R x =marginal revenu