3. Find the parametric equations of the of the curve \( c \), where \( c \) is the line segment from \( (1,2) \) to \( (4,7) \). A. \( x=1-5 t, y=2-9 t \) B. \( x=1-3 t, y=2-5 t \) C. \( x=1+3 t, y=2+5 t \) D. \( x=1+5 t, y=2+9 t \) 4. Evaluate \( \int_{c}(x) d s \), where \( c \) is the right half of the unit circle \( x^{2}+y^{2}=1 \). A. 1 B. 2 C \( 10 N \) D. \( \pi \) D. \( 4 \pi \)

To find the parametric equations for the line segment from \( (1,2) \) to \( (4,7) \), you can express \( x \) and \( y \) in terms of a parameter \( t \). Starting at \( (1,2) \) when \( t=0 \) and ending at \( (4,7) \) when \( t=1 \), the equations can be linearly interpolated, yielding \( x = 1 + 3t \) and \( y = 2 + 5t \). So, the answer is C: \( x=1+3 t, y=2+5 t \). Now, regarding the integral of the right half of the unit circle, you're essentially calculating the arc length. The right half of the unit circle can be parametrized as \( x = \cos(t) \) and \( y = \sin(t) \) where \( t \) ranges from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). Evaluating \( \int_{c}(x) d s \) ultimately turns out to be \( \pi \). Hence, the answer is D: \( \pi \).