Question

Let $(a,b)$ and $(\lambda ,\mu )$be two points on the curve $y=f\left(x\right).$ If the slope of the tangent to the curve at $(x,y)$ be $\varphi \left(x\right)$ then ${\int }_a^{\lambda }\varphi \left(x\right)dx$ is

• A. $\lambda -a$
• B. $\mu -b$
• C. $\lambda +\mu -a-b$
• D. none of these

Answer 1

The correct option is B $\mu -b$

$(a,b)$ and $(\lambda ,\mu )$ are two points on the curve $y=f\left(x\right)$
then $f\left(a\right)=b$ and $f\left(\lambda \right)=\mu$
$\varphi \left(x\right)=f^{\prime }\left(x\right)$
now, ${\int }_a^{\lambda }\varphi \left(x\right)dx={\int }_a^{\lambda }{f}^{\prime }\left(x\right)dx=f\left(\lambda \right)-f\left(a\right)=\mu -b$