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51.

Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point.

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52. Find the slope of the tangent to the curve x = at², y = 2at at t = 1.

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53.

Find the slope of the tangent to the curve y = x³ – x at x = 2.

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54.

Find the slope of the tangent to the curve y = 3x⁴ – 4x at (i) x = 1 (ii) x = 4.

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55.

Find the slope of the tangent to the curve $straight y space equals fraction numerator straight x minus 1 over denominator straight x minus 2 end fraction comma space space space straight x not equal to space 2 space space at space straight x space equals space 10$ .

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56. For the curve y = 3x² + 4 x , find the slope of the tangent to the curve at the point whose x-coordrnate is – 2.

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57. Find the slope of the tangent to the curve y = x³ – x + 1 at the point whose x-coordmate is 2.

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58.

Find the slope of the tangent to the curve y = x³ – 3x + 2 at the point whose. x-coordinate is 3.

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59.
Find the slope of the normal to the curve $straight x equals space straight a space cos cubed straight theta comma space space space straight y space equals space straight a space sin cubed straight theta space space space at space straight theta space equals space straight pi over 4$ .

straight x equals space straight a space cos cubed straight theta comma space space space straight y space equals space straight a space sin cubed straight theta space space space at space straight theta space equals space straight pi over 4

The equation of curve are

straight x space equals space straight a space cos cubed straight theta comma space space space space space straight y equals space straight a space sin cubed straight theta space

therefore space space space space space space space dx over dθ space equals space minus 3 straight a space cos squared straight theta space space sin space straight theta comma space space space space dy over dθ space equals space 3 straight a space space sin squared straight theta space cos space straight theta space space space space space space space space space space space space space space dy over dx space equals space fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction space equals space fraction numerator 3 straight a space sin squared straight theta space space cos space straight theta over denominator negative 3 space straight a space cos squared straight theta space sin space straight theta end fraction space equals negative fraction numerator sin space straight theta over denominator cos space straight theta end fraction space equals space minus tan space straight theta comma which space is space slope space of space tangent space to space the space curve

therefore space space space space space slope space of space normal space to space the space curve space equals space minus fraction numerator 1 over denominator begin display style dy over dx end style end fraction space equals space minus fraction numerator 1 over denominator negative tan space straight theta end fraction space equals space cot space straight theta At space space space straight theta space equals space straight pi over 4 comma space space space space slope space of space normal space to space the space curve space equals space cot space straight pi over 4 space equals space 1

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60.

Find the slope of the normal to the curve $straight x space equals space 1 minus straight a space sin space straight theta comma space space space straight y space equals space space straight b space cos squared straight theta space space at space straight theta space equals space straight pi over 2.$

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59. Find the slope of the normal to the curve .

59.
Find the slope of the normal to the curve $straight x equals space straight a space cos cubed straight theta comma space space space straight y space equals space straight a space sin cubed straight theta space space space at space straight theta space equals space straight pi over 4$ .