Tangents, Normals And Rates Of Change
Bayani Gaba-gaba
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.
Manufura
- 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
Takardar Darasi
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- 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
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