Winter wk 7 – Tues.15.Feb.05 Antiderivative = integral = area under curve Derivative = slope of curve Review 6.1-2: Finding antiderivatives 6.3: Introduction.

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Winter wk 7 – Tues.15.Feb.05 Antiderivative = integral = area under curve Derivative = slope of curve Review 6.1-2: Finding antiderivatives 6.3: Introduction to differential equations 6.4: Second fundamental theorem of calculus 6.5: Equations of motion Energy Systems, EJZ

Review Ch.6.1-2: Antiderivatives Recall: if F=slope=df/dx =f’, then Given a graph of the slope F=df/dx, sketch f: Where is slope F=0? There f is flat. Where is slope F>0? There f increases. Where is slope F<0? There f decreases. f(x)=AREA under F(x) curve.

Analytic antiderivatives Polynomials: Exponential and logarithmic functions: Trigonometric functions:

6.3 Introduction to DiffEq Speed = rate of change of position v = ds/dt This is a differential equation! Do Conceptest 1, then solve for s if v=50 Acceleration = rate of change of speed a = dv/dt Solve for v and s if a = -g (gravity) See Ex.2 and 3, then do Conceptests 2&3.

Conceptest 1 Graphs (I)-(III) show velocity versus time for three different objects. Which travels the furthest in four seconds? Which travels the shortest distance?

Conceptest 1 answer

Conceptest 2

Conceptest 2 answer

Conceptest 3

Conceptest 3 answer Practice Ch.6.3 # odd-numbered probs thru #9; 10, 12, 18, 20, 22

6.4: Second Fundamental theorem Definite integrals have limits specified, including the intercept. Indefinite integrals have no limits specified, so the intercept is unknown.

Conceptest 4

Conceptest 4 answer

Conceptest 5

Conceptest 5 answer

Conceptest 6

Conceptest 6 answer Practice Ch.6.4 # 1, 2, 6, 8, 10, 12, 14, 18, 20

6.5 Equations of Motion You found that if a = dv/dt = -g, then v = -gt + C 1. If the initial velocity v( t=0 ) = v 0, then find C 1 : v = -gt + _____ Displacement s = -½ gt 2 + v 0 t + C 2 If the initial position s( t=0 ) = s 0, then find C 2 : s = __________________

History and Forces These simple equations of motion were sought for 2000 years and finally discovered by Newton, who invented calculus for this purpose. Forces can accelerate masses: F=ma AND masses in motion tend to stay in motion: Inertia Ex: Recall the forces you know, and derive their potential energies. Practice Ch.6.5 # odd-numbered probs thru #35; 36, 58