C. The Instantaneous Rate of Change of one variable, Q, with respect to another variable t...

 
 

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C. The Instantaneous Rate of Change of one variable, Q, with respect to another variable t is denoted dQ dt and requires only ONE value of t, since you are calculating the rate of change at a single instant. You are actually calculating a slope of a Tangent Line which is really a derivative. See example Il below for several examples. Returning to Example 1, we have been examining the average rate of change of Q wrt t from t = 1 to other times that are getting closer and closer to 1. If we take the limit as the second time gets closer and closer to t = 1 we will obtain the INSTANTANEOUS rate of change of Q wrt t @t=1. This is the DERIVATIVE of Q wrt t evaluated at t = 1. a) So the instantaneous rate of change of Q wrt t @ t = 1 is denoted: 1. C) dQ dt Itt which stands for the derivative of Q wrt t evaluated @t=1. Since X(PLORE) gives the derivative of t2 + 3t-1 as 2t + 3, we can find dQ dit a) = = (2t+3), = 2(1)+3=5 b) Find the instantaneous rate of change of Q wrt t @ t = 2, i.e. -2. Ans. 7 (1000 bact./hr) (1000 bacteria per hour). dQ dit Find the instantaneous rate of change of Q wrt t @ t = 3, i.e. bact./hr) 105 1-2t Example II Suppose A represents the amount of a drug (in milligrams) left in the body t hours after being injected intravenously. Suppose A = : Use X(PLORE) to find the derivative. dA milligrams. NOTE: dt dt 3 Ans. (1000 -210 (1+2t)-² Find the average rate of change of A wrt t over the time interval of t = 1 to t = 3. Ans. mg/hr b) Find the instantaneous rate of change of A wrt t at t = 2. c) D. Rates of change in PHYSICS Example III. RECTILINEAR MOTION --motion along a straight line. In what follows we will have a particle moving along a straight line which contains a REFERENCE point called O and a positive direction indicated. We will usually use a horizontal line and have the positive direction on the right side. Find the instantaneous rate of change of A wrt t at t = 1. As At = b) c) d) e) In addition, s will represent the signed distance (in meters) from the moving particle to the reference point. Also t will represent the time at which we locate the particle with t given in seconds. Ans. Suppose s = 2t³ - 4t² + 2t - 1 (in meters) .RECALL:Use table for average rate of change and DERIVATIVE for instantaneous rate of change. a) What is the average rate of change of s wrt t (called AVERAGE VELOCITY) over the time interval from t = 1 tot = 3? As m/sec At m/sec m/sec Ans. ds dt lt=2== mg/hr m/sec mg/hr ds dit 11=1 = What is the average rate of change of s wrt t (called AVERAGE VELOCITY) over the time interval from t = 2 tot = 4? What is the instantaneous rate of change of s wrt t (called INSTANTANEOUS VELOCITY and denoted by the letter v) at the time t = 2? What is the instantaneous rate of change of s wrt t (called INSTANTANEOUS VELOCITY) at the time t = 1? At what time is the particle at rest? i.e. when is the instantaneous velocity equal to 0? Sec C. The Instantaneous Rate of Change of one variable, Q, with respect to another variable t is denoted dQ dt and requires only ONE value of t, since you are calculating the rate of change at a single instant. You are actually calculating a slope of a Tangent Line which is really a derivative. See example Il below for several examples. Returning to Example 1, we have been examining the average rate of change of Q wrt t from t = 1 to other times that are getting closer and closer to 1. If we take the limit as the second time gets closer and closer to t = 1 we will obtain the INSTANTANEOUS rate of change of Q wrt t @t=1. This is the DERIVATIVE of Q wrt t evaluated at t = 1. a) So the instantaneous rate of change of Q wrt t @ t = 1 is denoted: 1. C) dQ dt Itt which stands for the derivative of Q wrt t evaluated @t=1. Since X(PLORE) gives the derivative of t2 + 3t-1 as 2t + 3, we can find dQ dit a) = = (2t+3), = 2(1)+3=5 b) Find the instantaneous rate of change of Q wrt t @ t = 2, i.e. -2. Ans. 7 (1000 bact./hr) (1000 bacteria per hour). dQ dit Find the instantaneous rate of change of Q wrt t @ t = 3, i.e. bact./hr) 105 1-2t Example II Suppose A represents the amount of a drug (in milligrams) left in the body t hours after being injected intravenously. Suppose A = : Use X(PLORE) to find the derivative. dA milligrams. NOTE: dt dt 3 Ans. (1000 -210 (1+2t)-² Find the average rate of change of A wrt t over the time interval of t = 1 to t = 3. Ans. mg/hr b) Find the instantaneous rate of change of A wrt t at t = 2. c) D. Rates of change in PHYSICS Example III. RECTILINEAR MOTION --motion along a straight line. In what follows we will have a particle moving along a straight line which contains a REFERENCE point called O and a positive direction indicated. We will usually use a horizontal line and have the positive direction on the right side. Find the instantaneous rate of change of A wrt t at t = 1. As At = b) c) d) e) In addition, s will represent the signed distance (in meters) from the moving particle to the reference point. Also t will represent the time at which we locate the particle with t given in seconds. Ans. Suppose s = 2t³ - 4t² + 2t - 1 (in meters) .RECALL:Use table for average rate of change and DERIVATIVE for instantaneous rate of change. a) What is the average rate of change of s wrt t (called AVERAGE VELOCITY) over the time interval from t = 1 tot = 3? As m/sec At m/sec m/sec Ans. ds dt lt=2== mg/hr m/sec mg/hr ds dit 11=1 = What is the average rate of change of s wrt t (called AVERAGE VELOCITY) over the time interval from t = 2 tot = 4? What is the instantaneous rate of change of s wrt t (called INSTANTANEOUS VELOCITY and denoted by the letter v) at the time t = 2? What is the instantaneous rate of change of s wrt t (called INSTANTANEOUS VELOCITY) at the time t = 1? At what time is the particle at rest? i.e. when is the instantaneous velocity equal to 0? Sec

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