Tangents, Normals And Rates Of Change
Gbogbo ọrọ náà
A tangent line just touches a curve at a single point, running in exactly the same direction as the curve at that instant. Its perpendicular partner, the normal, points straight into the curve. Together they unlock a family of problems that link algebra, coordinate geometry and calculus in a single question.
In this topic you will learn how to use differentiation to find the gradient of a curve at any point, build the equations of tangent and normal lines, and extend the chain rule to connected rates of change. You will also meet the small-change approximation, a powerful tool for estimating how a function responds to a tiny shift in its input.
Ebumnobi
- Find the equations of tangents and normals to the curve y = f(x)
- Apply calculus to rates of change and connected rates of change
- Understand and use the approximation dy is approximately equal to (dy/dx) times dx for small dx
Akọmọ Ojú-ẹkọ
Ọ dị na ngwa Green Bridge
Budata ngwa Green Bridge CBT na ekwentị maọbụ kọmputa gị iji nweta akwụkwọ ndụmọdụ zuru oke, ajụjụ mmụta, na ndị ọzọ.
Ayẹwo Ẹkọ
Ekele diri gi maka imecha ihe karịrị na Tangents, Normals And Rates Of Change. Ugbu a na ị na-enyochakwa isi echiche na echiche ndị dị mkpa, ọ bụ oge iji nwalee ihe ị ma. Ngwa a na-enye ụdị ajụjụ ọmụmụ dị iche iche emebere iji kwado nghọta gị wee nyere gị aka ịmata otú ị ghọtara ihe ndị a kụziri.
Ị ga-ahụ ngwakọta nke ụdị ajụjụ dị iche iche, gụnyere ajụjụ chọrọ ịhọrọ otu n’ime ọtụtụ azịza, ajụjụ chọrọ mkpirisi azịza, na ajụjụ ede ede. A na-arụpụta ajụjụ ọ bụla nke ọma iji nwalee akụkụ dị iche iche nke ihe ọmụma gị na nkà nke ịtụgharị uche.
Jiri akụkụ a nke nyocha ka ohere iji kụziere ihe ị matara banyere isiokwu ahụ ma chọpụta ebe ọ bụla ị nwere ike ịchọ ọmụmụ ihe ọzọ. Ekwela ka nsogbu ọ bụla ị na-eche ihu mee ka ị daa mba; kama, lee ha anya dị ka ohere maka ịzụlite onwe gị na imeziwanye.
- The curve y = x^2 + 3x passes through (1, 4). What is the gradient of the tangent at this point? A) 4 B) 5 C) 6 D) 7 Answer: B
- If the tangent to a curve at a point has gradient 4, what is the gradient of the normal? A) 4 B) -4 C) 1/4 D) -1/4 Answer: D
- The radius of a circle increases at 2 cm/s. Given A = pi r^2, what is dA/dt when r = 5? A) 10 pi B) 20 pi C) 25 pi D) 50 pi Answer: B
- For y = x^3, the small-change approximation gives delta y approximately equal to 3x^2 delta x. If x = 2 and delta x = 0.01, what is delta y approximately? A) 0.06 B) 0.12 C) 0.24 D) 0.36 Answer: B
- At a point on a curve, dy/dx = 0. What can you say about the normal at this point? A) It is horizontal B) It is vertical C) It has gradient 1 D) It does not exist Answer: B
Ọ dị na ngwa Green Bridge
Budata ngwa Green Bridge CBT na ekwentị maọbụ kọmputa gị iji nweta akwụkwọ ndụmọdụ zuru oke, ajụjụ mmụta, na ndị ọzọ.
Ọ dị na ngwa Green Bridge
Budata ngwa Green Bridge CBT na ekwentị maọbụ kọmputa gị iji nweta akwụkwọ ndụmọdụ zuru oke, ajụjụ mmụta, na ndị ọzọ.
Àwọn Ìwé Tó Dáa Láti Kà
Meecha Ajụjụ Ule Ọmarịcha
Ị chọrọ ime ajụjụ ule ọmarịcha gbasara Tangents, Normals And Rates Of Change? Budata ngwa Green Bridge CBT iji nweta ajụjụ ule ọmarịcha na nyocha zuru ezu gbasara isiokwu a.
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