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VICMath MethodsQuick questions

Unit 3

Quick questions on Tangents, stationary points and curve sketching: VCE Math Methods Unit 3

10short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is tangent line?
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At the point (a,f(a))(a, f(a)) on y=f(x)y = f(x), the tangent line has slope fβ€²(a)f'(a) and passes through (a,f(a))(a, f(a)). In point-slope form:
What is normal line?
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The normal is perpendicular to the tangent at the same point, so its slope is βˆ’1fβ€²(a)-\frac{1}{f'(a)} (provided fβ€²(a)β‰ 0f'(a) \neq 0):
What are intercepts?
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y-intercept: f(0)=0f(0) = 0. x-intercepts: factor x3βˆ’6x2+9x=x(x2βˆ’6x+9)=x(xβˆ’3)2x^3 - 6 x^2 + 9 x = x(x^2 - 6 x + 9) = x(x - 3)^2. Zeros at x=0x = 0 (single root) and x=3x = 3 (double root).
What are stationary points?
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fβ€²(x)=3x2βˆ’12x+9=3(x2βˆ’4x+3)=3(xβˆ’1)(xβˆ’3)f'(x) = 3 x^2 - 12 x + 9 = 3(x^2 - 4 x + 3) = 3 (x - 1)(x - 3). Stationary at x=1x = 1 and x=3x = 3.
What is point of inflection?
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fβ€²β€²(x)=0f''(x) = 0 at x=2x = 2. f(2)=8βˆ’24+18=2f(2) = 8 - 24 + 18 = 2. Inflection at (2,2)(2, 2).
What is end behaviour?
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Cubic with positive leading coefficient: fβ†’βˆ’βˆžf \to -\infty as xβ†’βˆ’βˆžx \to -\infty, fβ†’+∞f \to +\infty as xβ†’+∞x \to +\infty.
What is wrong slope for the normal?
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The normal slope is βˆ’1fβ€²(a)-\frac{1}{f'(a)}, not βˆ’fβ€²(a)-f'(a) or 1fβ€²(a)\frac{1}{f'(a)}.
What is q1?
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Find the equation of the tangent to y=x2+1y = x^2 + 1 at x=2x = 2. [3 marks]
What is q2?
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Find and classify the stationary points of f(x)=x3βˆ’12xf(x) = x^3 - 12x. [4 marks]
What is q3?
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Find the point of inflection of f(x)=x3βˆ’3x2+2f(x) = x^3 - 3x^2 + 2. [3 marks]

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