Application of Derivatives

Prove that the line is a tangent to the curve at the point where the curve cuts y-axis. - Wired Faculty

Question

Prove that the line $straight x over straight a plus straight y over straight b space equals space 1$ is a tangent to the curve $straight y space equals be to the power of negative straight x over straight a end exponent$ at the point where the curve cuts y-axis.

Solution

The equation of given curve is $straight y space equals space be to the power of negative straight x over straight a end exponent$ ...(1)
It cuts y-axis where x = 0
$therefore space space space space space space putting space straight x space equals space 0 space in space left parenthesis 1 right parenthesis comma space we space get comma space space space space space space space space space space space space straight y equals be to the power of 0 space equals space straight b cross times 1 space equals space straight b therefore space space space space space curve space left parenthesis 1 right parenthesis space cuts space straight y minus axis space at space left parenthesis 0 comma space straight b right parenthesis.$
Differentiating (1), w.r.t. x, we get,
$dy over dx space equals space minus straight b over straight a straight e to the power of negative straight x over straight a end exponent$
$At space left parenthesis 0 comma space straight b right parenthesis comma space space dy over dx space equals space minus straight b over straight a straight e to the power of 0 space equals space minus straight b over straight a cross times 1 space equals space minus straight b over straight a therefore space space space space straight m space equals space minus straight b over straight a$
The equation of tangent (0, b) with $straight m space equals space minus straight b over straight a space space is$
$straight y minus straight b space equals space minus straight b over straight a left parenthesis straight x minus 0 right parenthesis$ $open square brackets straight y minus straight y subscript 1 space equals space straight m left parenthesis straight x minus straight x subscript 1 right parenthesis close square brackets$
$space space space space space ay minus ab space equals space minus bx comma space space space space space or space space space bx plus ay space equals space ab space or space space space space space straight x over straight a plus straight y over straight b space space equals space 1 Hence space the space result. space$

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Application Of Derivatives

Prove that the line $straight x over straight a plus straight y over straight b space equals space 1$ is a tangent to the curve $straight y space equals be to the power of negative straight x over straight a end exponent$ at the point where the curve cuts y-axis.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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