= h. Geometrically this quantity represents the slope of the secant line connecting the points
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1 Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f ( )) and, f ( )). ( ( Pysically te average rate of cange is te velocity. Instantaneous rate of cange: Te rate of cange of y f() wit respect to at is y lim y lim y lim 0 ) ) lim 0 ) Tis is also called te derivative at f ( ) lim 0 ) Pysics: Geometrically tis is te slope of te tangent line at Pysically te rate of cange is te instantaneous velocity at. Linear motion/ Free fall: Position Function: s( is te position function of a particle tat is moving in a straigt line. Velocity Function: Te velocity is te cange in displacement s wit respect to time ds v ( s ( dt Interpretation of te velocity: Since te velocity represents te slope of te position function it tells us ow fast and in wic direction te position is canging.. If v( > 0 ten te grap of s( is increasing and te position is increasing: i.e. te object is moving to te rigt/up. If v( < 0 ten te grap of s( is decreasing and te position is decreasing: i.e. te object is moving to te left/down.
2 dv d s Acceleration Function: a ( s ( v ( dt dt Interpretation of te acceleration: Te acceleration represents te slope (rate of cange) of te velocity.. Wen a ( > 0 ten te velocity is increasing. Wen a ( < 0 ten te velocity is decreasing if v( > 0 te object is speeding up if v( < 0 te object is slowing up if v( > 0 te object slows down if v( < 0 te object speeds up So wen te velocity and acceleration are acting in te same direction (ave te same algebraic sign) te object speeds up. Q. Wat is te difference between speed and velocity? Eample:. Suppose s ( t 4t 3 gives te position of a body moving on a coordinate line for 0 t 6 were t is measured in seconds and s in meters. a) Determine te position of te particle at te endpoints. b) Wat is te displacement? c) Determine te average velocity over te interval [0, 6] d) Determine te velocity after s and after 4s.
3 e) Determine wen te particle canges direction and wen te particle is at rest. f) Wen te particle is moving forward? (i.e. in te positive direction.) g) Find te total distance traveled by te particle on te interval [0, 6]. ) Draw a diagram to represent te motion of te particle.. A projectile is fired straigt up from te ground. Its distance from te ground after t seconds is s( 6t + 00t (measured in fee a) Wat is te velocity at t 4 sec and at t 9 sec? b) Wen does te projectile reaces its maimum eigt? c) Find te maimum eigt of te projectile?
4 d) Wen does te projectile it te ground? e) Find te impact velocity. Linear Density, ρ If te mass of a rod or piece of wire can be measured from its left end to a point on te rod as m f () ten we can find te mass of te rod tat lies between and by applying te formula m ) ). Similarly we can find te average density over tis region by using our average formula: average density m ) ) Te linear density ρ at is te rate of cange of mass wit respect to lengt. I.e. te linear density is te derivative of mass wit respect to lengt: dm ρ d If te rod appens to be omogeneous (i.e. as constant density) ten te linear density is uniform and is defined as te mass per unit lengt ( ρ m / l ) and is measured in kilograms per meter. Eample: 3. Te mass of a part of a rod is given by m, were is measured in meters and m in kilograms. a) Find te average density of te part of te rod given by. b) Find te linear density at.
5 4. A rock is tossed straigt up from a cliff ft above te ground. Its distance from te ground below after t seconds is s ( 6t + 96t + (measured in fee. a) Determine te velocity and te acceleration functions. b) Wat is te initial velocity of te rock? c) Find te maimum eigt of te rock. d) Determine te rocks impact velocity. 5. A sperical balloon is being inflated. Find te rate of increase of te surface area ( S wit respect to te radius r were r is ft. 4πr )
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