Notes
Note Matlab carefully the use of ( ) and [ ].
Velocity, Tangents & Rates of Change
Lecture 5
h=0.5; t=1:h:5; s=t.^2; What is the sixth value of s ? s(6) How to nd s(6)-s(5), s(5)-s(4), s(4)-s(3), s(3)-s(2), s(2)-s(1) all in one go? That is, we want s([2 3 4 5 6])-s([1 2 3 4 5]) So use s(2:6)-s(1:5) Try it! Or, neater still: t1=2:6; t2=1:5; s(t1)-s(t2)
Algebra & Calculus I
Calculus:
Velocity, Tangents & Rates of Change
Lect. 5
1 / 13
Calculus:
Velocity, Tangents & Rates of Change
Lect. 5
4 / 13
The Baseball Throwup
Velocity is a vector, v, speed is its length |v|.
Actual Velocity at t = 0.4: a First Approximation?
We need more data! Position s in metres, time t in seconds, velocity in m/s. t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s 0 1.8 3.2 4.2 4.8 5 4.8 4.2 3.2 1.8 0 Choose a direction for velocity, say, positive up. Average velocity from t = 1.2 to 1.4 ? (Answer: -3) Get data for smaller step size h = t = 0.01 t 0.39 0.4 0.41 s 3.1395 3.2 3.2595 Average velocity from t = 0.39 to 0.4 ? (Answer: 6.05) Average velocity from t = 0.4 to 0.41 ? (Answer: 5.95)
Calculus:
Velocity, Tangents & Rates of Change
Lect. 5
6 / 13
Calculus:
Velocity, Tangents & Rates of Change
Lect. 5
7 / 13
Actual Velocity at t = 0.4: a Better Approximation?
Actual Velocity at t = 0.4: Exact Value?
Actual velocity at t = 0.4 appears to be 6.
We still need more data! Get data for even smaller step size h = t = 0.001 t 0.399 0.4 0.401 s 3.193995 3.205995 3.2 Average velocity from t = 0.399 to 0.4 ? (Answer: 6.004999) Average velocity from t = 0.4 to 0.401 ? (Answer: 5.994999)
Average velocity from t = 0.4 to 0.4 + h is dened to be s(0.4 + h) s(0.4) h i.e. position at t = 0.4 + h minus position at 0.4 divided by time step h Instantaneous velocity at t = 0.4 dened as s(0.4 + h) s(0.4) h0 h lim It is the slope of the tangent to the s versus t curve at t = 0.4.
Calculus:
Velocity, Tangents & Rates of Change
Lect. 5
8 / 13
Calculus:
Velocity, Tangents & Rates of Change
Lect. 5
9 / 13
Velocity as Slope of a Curve
Summary: Velocity as a Limit
Note: the (delta) notation for a change or increment (Stewart 2.6).
Exercise
To see the curve of the above baseball heights, plot the curve for s = 10t 5t2 from t = 0 to 2 . Measure the slope at t = 0.4 (you could use zoom in Matlab ). It should be 6.
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