Find the second derivative of the vector-valued function →r(t)=(5t+7sin⁡(t))i+(7t+2cos⁡(t))j

Question

Find the second derivative of the vector-valued function  →r(t)=(5t+7sin⁡(t))i+(7t+2cos⁡(t))j

Befoodly · Accepted Answer

Step 1  Given that:  r(t)=(5t+7sin⁡t)i+(7t+2cos⁡t)j  So, 1st derivative of r:  drdt=iddt(5t+7×sin⁡t)+j×ddt(7t+2cos⁡t)  =i(5ddt(t)+7ddt(sin⁡t))+j(7ddt(t)+2ddtcos⁡t)  =i(5+7cos⁡t)+j(7−2sin⁡t)  So, drdt=(5+7cos⁡t)i+(7−2sin⁡t)j  2nd derivative of r:  d2rdt2=ddt[(5+7cos⁡t)i+(7−2sin⁡t)j]  =(5ddt(1)+7ddtcos⁡t)i+(7ddt(1)−2ddtsin⁡t)j  =(0−7sin⁡t)i+(0−2cos⁡t)j  −7sin⁡t×i−2cos⁡t×j  So, d2rdt2=−7sin⁡t×i−2cos⁡t×j

Find the second derivative of the vector-valued function →r(t)=(5t+7sin⁡(t))i+(7t+2cos⁡(t))j

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