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Rog_Sec_13_2-13_3-11_2

Rog_Sec_13_2-13_3-11_2 - Math 20C Lecture Examples Sections...

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(8/3/08) Math 20C. Lecture Examples. Sections 11.2, 13.2, and 13.3. Calculus of vector-valued functions Example 1 Find lim t 1 ( t 2 - 3 , e 3 t , ln t ) . Answer: lim t 1 ( t 2 - 3 , e 3 t , ln t ) = (- 2 , e 3 , 0 ) Example 2 What is lim t 3 r ( t ) if r ( t ) = (- t, t 2 - 5 ) ? Answer: lim t 3 r ( t ) = (- 3 , 4 ) Example 3 Find the derivative d dt ( t 2 - 3 , e 3 t , ln t ) . Answer: d dt ( t 2 - 3 , e 3 t , ln t ) = (big 2 t, 3 e 3 t , 1 t )big Example 4 What is the derivative r prime ( 1 3 π ) for r ( t ) = 2 cos t i + 4 sin t j ? Answer: r prime ( 1 3 π ) = - 3 i + 2 j Velocity vectors and speed Suppose that an object moving in an xy -plane has position vector r = r ( t ) at time t (Figure 1). For small positive Δ t , the displacement vector r ( t + Δ t ) - r ( t ) points from one point to the other on the curve in the direction of the object’s motion. Dividing r ( t + Δ t ) - r ( t ) by the positive number Δ t to form the difference quotient, r ( t + Δ t ) - r ( t ) Δ t (1) changes the length of the vector but not its direction (Figure 2). This vector also lies along a secant line and points in the direction of the object’s motion for Δ t < 0 because then r ( t + Δ t ) - r ( t ) points in the opposite direction and dividing it by a negative number reverses its direction. FIGURE 1 FIGURE 2 Suppose that the derivative r prime ( t ) exists. Then r = r ( t ) is continuous at t and the tip of r ( t + Δ t ) approaches the tip of r ( t ) as Δ t 0. Moreover, if the derivative is not the zero vector, then the secant line approaches a line, through the tip of r ( t ) and parallel to r prime ( t ). We define the line to be the tangent line to the curve and refer to r prime ( t ) as the velocity vector v ( t ) of the moving object at that point (Figure 3). We also define the length of the velocity vector to be the object’s
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