The figure shows a curve $C$ given by a vector function $\mathbf { r } ( t )$ . Choose the correct expression for $\mathbf { r } ^ { \prime } ( 4 )$ .
A) $\lim _ { h \rightarrow 0 } \frac { r ( 4 - h ) + r ( 4 ) } { h }$
B) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) - r ( 4 ) } { h }$
C) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) - r ( h ) } { h }$
D) $\lim _ { h \rightarrow 0 } \frac { r ( 4 - h ) - r ( 4 ) } { h }$
E) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) + r ( 4 ) } { h }$

Question

The figure shows a curve $C$ given by a vector function $\mathbf { r } ( t )$ . Choose the correct expression for $\mathbf { r } ^ { \prime } ( 4 )$ . 
A) $\lim _ { h \rightarrow 0 } \frac { r ( 4 - h ) + r ( 4 ) } { h }$ 
B) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) - r ( 4 ) } { h }$ 
C) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) - r ( h ) } { h }$ 
D) $\lim _ { h \rightarrow 0 } \frac { r ( 4 - h ) - r ( 4 ) } { h }$ 
E) $\lim _ { h \rightarrow 0 } \frac { r ( 4 + h ) + r ( 4 ) } { h }$

Quizplus · Accepted Answer

The expression $\mathbf{r}'(4)$ represents the derivative of the vector function $\mathbf{r}(t)$ evaluated at $t = 4$. Using the limit definition of the derivative, we have $$\mathbf{r}'(4) = \lim_{h \to 0} \frac{\mathbf{r}(4 + h) - \mathbf{r}(4)}{h}.$$ This matches choice (B).

The Figure Shows a Curve CCC Given by a Vector Function r(t)\mathbf { r } ( t )r(t)

The Figure Shows a Curve $C$ Given by a Vector Function $\mathbf { r } ( t )$