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Application of Derivatives

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Short Answer Type

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.

The equation of curve is
$straight y space equals space 3 straight x to the power of 4 minus 4 straight x$
$therefore space space space space dy over dx space equals space 12 straight x cubed minus 4 comma space space which space is space slope space of space tangent space to space the space curve.$
(i) At x = 1, $dy over dx space equals space 12 left parenthesis 1 right parenthesis cubed space minus space 4 space equals space 12 space minus 4 space space equals space 8$
which is required slope of tangent to the curve.
(ii) $At space straight x space equals space 4 comma space space space dy over dx space equals space 12 left parenthesis 4 right parenthesis cubed space minus space 4 space equals space 12 space cross times space 64 space minus space 4 space equals space 768 space minus space 4 space equals space space 764$
which is required slope of tangent to the curve

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

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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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Previous Year Papers

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Short Answer Type

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

54.
Find the slope of the tangent to the curve y = 3x⁴ – 4x at (i) x = 1 (ii) x = 4.