a train moves along a straight line. its location at time t is given by ( s(t)=\frac{50}{t}, 1 leq t leq 9…

a train moves along a straight line. its location at time t is given by ( s(t)=\frac{50}{t}, 1 leq t leq 9 ), where t is measured in hours and ( s(t) ) is measured in kilometers. complete parts (a) - (c) below. where on the graph of ( s(t) ) can you find the instantaneous velocity? a. the instantaneous velocity can be seen on the graph as the slope of the secant line joining the two points ( (1, s(1)) ) and ( (3, s(3)) ). b. the instantaneous velocity can be seen on the graph as the slope of the secant line joining the two points ( (1, s(1)) ) and ( (9, s(9)) ). c. the instantaneous velocity can be seen on the graph as the slope of the tangent line to the curve at the point ( (2, s(2)) ). d. the instantaneous velocity can be seen on the graph as the value of the function at ( t = 2 ).

a train moves along a straight line. its location at time t is given by ( s(t)=\frac{50}{t}, 1 leq t leq 9 ), where t is measured in hours and ( s(t) ) is measured in kilometers. complete parts (a) - (c) below. where on the graph of ( s(t) ) can you find the instantaneous velocity? a. the instantaneous velocity can be seen on the graph as the slope of the secant line joining the two points ( (1, s(1)) ) and ( (3, s(3)) ). b. the instantaneous velocity can be seen on the graph as the slope of the secant line joining the two points ( (1, s(1)) ) and ( (9, s(9)) ). c. the instantaneous velocity can be seen on the graph as the slope of the tangent line to the curve at the point ( (2, s(2)) ). d. the instantaneous velocity can be seen on the graph as the value of the function at ( t = 2 ).

Answer

Brief Explanations:

Recall the definition of instantaneous velocity. The instantaneous velocity of a function (s(t)) at a point (t = a) is given by the derivative (s^{\prime}(a)). Geometrically, the derivative of a function (y = f(x)) at a point (x=a) is the slope of the tangent line to the curve (y = f(x)) at the point ((a,f(a))).

  • Option A: The slope of the secant line joining ((1,s(1))) and ((3,s(3))) gives the average velocity over the interval ([1,3]), not the instantaneous velocity.
  • Option B: The slope of the secant line joining ((1,s(1))) and ((9,s(9))) gives the average velocity over the interval ([1,9]), not the instantaneous velocity.
  • Option C: Since the instantaneous velocity at (t = 2) is given by (s^{\prime}(2)), and geometrically (s^{\prime}(2)) is the slope of the tangent line to the curve (s(t)) at the point ((2,s(2))).
  • Option D: The value of the function (s(t)) at (t = 2) (i.e., (s(2)=\frac{50}{2}=25)) gives the position of the train at (t = 2) hours, not the instantaneous velocity.

Answer:

C. The instantaneous velocity can be seen on the graph as the slope of the tangent line to the curve at the point ((2, s(2))).