Sketch the curve y = x^3 . (a) Find the equation of the tangent to the curve at A(1,1). (b) Find the coordinates of point B, where the tangent meets the curve again. (c) Calculate the area between the tangent B and the arc AB of the curve.


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Question Number 208052 by necx122 last updated on 03/Jun/24

$S k e t c h t h e c u r v e y = x^{3} .$ $(a) F i n d t h e e q u a t i o n o f t h e t a n g e n t$ $t o t h e c u r v e a t A (1, 1) .$ $(b) F i n d t h e c o o r d i n a t e s o f p o i n t B,$ $w h e r e t h e t a n g e n t m e e t s t h e c u r v e a g a i n .$ $(c) C a l c u l a t e t h e a r e a b e t w e e n t h e$ $t a n g e n t B a n d t h e a r c A B o f t h e c u r v e .$
Commented by Frix last updated on 03/Jun/24

$Tangent t : y = a x + b$ $f : y = x^{3} \Rightarrow f^{'} : y = 3 x^{2}$ $P \in f : P = (\begin{matrix} p \\ p^{3} \end{matrix})$ $P \in t : p^{3} = a p + b but a = 3 p^{2} \Rightarrow b = - 2 p^{3} \Rightarrow$ $t : y = 3 p^{2} x - 2 p^{3}$ $f (x) = x^{3}$ $t (x) = 3 p^{2} x - 2 p^{3}$ $d (x) = f (x) - t (x) = x^{3} - 3 p^{2} x + 2 p^{3}$ $d (x) = {(x - p)}^{2} (x + 2 p)$ $This has a double zero at x = p and a single$ $zero at x = - 2 p . Its extreme is at x = - p$ $with d (- p) = 4 p^{3} \Rightarrow for p < 0 we have a$ $local \min, for p > 0 a local \max \Rightarrow$ $4 p^{3} ⩽ d (x) ⩽ 0; p < 0 \land p ⩽ x ⩽ - 2 p$ $\Rightarrow Area is - \underset{p}{\int^{- 2 p}} d (x) d x = \underset{- 2 p}{\int^{p}} d (x) d x$ $0 ⩽ d (x) ⩽ 4 p^{3}; p > 0 \land - 2 p ⩽ x ⩽ p$ $\Rightarrow A rea is \underset{- 2 p}{\int^{p}} d (x) d x$ $\Rightarrow Area for p \neq 0 is always \underset{- 2 p}{\int^{p}} d (x) d x$
Commented by Frix last updated on 03/Jun/24

$y = x^{3}$ $y^{'} = 3 x^{2}$ $Tangent at x = p : y = 3 p^{2} x - 2 p^{3}$ $Intersection$ $x^{3} - (3 p^{2} x - 2 p^{3}) = 0$ ${(x - p)}^{2} (x + 2 p) = 0$ $x_{1} = p (obvious)$ $x_{2} = - 2 p$ $The area is$ $\underset{- 2 p}{\int^{p}} (x^{3} - (3 p^{2} x - 2 p^{3})) d x = \frac{27 p^{4}}{4}$
Commented by necx122 last updated on 03/Jun/24

$T h a n k y o u s i r f r i x . H o w e v e r, I^{'} m$ $b o t h e r e d a b o u t h o w y o u c a m e a b o u t$ $t h e y - i n t e r c e p t f o r t h e t a n g e n t a s - 2 p^{3} .$ $A l s o, i n s k e t c h i n g t h e c u r v e a n d$ $c a l c u l a t i n g t h e a r e a b o u n d e d b y t h e$ $a r e n t w e s u p p o s e d t o c a l c u l a t e f o r a l l$ $p o s s i b l e r e g i o n s b o t h a b o v e a n d b e l o w$ $t h e x - a x i s w h i c h m a y i n d i c a t e a d i f f e r e n t$ $v a l u e f o r t h e a r e a b o u n d e d b y t h e c u r v e$ $a n d t h e l i n e .$ $T h a n k y o u f o r y o u r h e l p . I a l w a y s$ $a p p r e c i a t e .$
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