Solved: "Find the slope of the tangent line to..." - Plainmath

Felix Fitzgerald

Open question

2022-08-20

Find the slope of the tangent line to the parabola

y = 4 x - x^{2}

at the point (1,3), i) using Definition 1, ii) using Equation 2

Answer & Explanation

tkaljegs

Beginner2022-08-21Added 9 answers

Definition 2:

m = lim_{x \Rightarrow a} \frac{f (x) - f (a)}{x - a}

i)

m = lim_{x \Rightarrow 1} \frac{f (x) - f (1)}{x - 1}

= lim_{x \Rightarrow 1} \frac{(4 x - x^{2}) - (4 (1) - 1^{2})}{x - 1}

= lim_{x \Rightarrow 1} \frac{- x^{2} + 4 x - 3}{x - 1}

= lim_{x \Rightarrow 1} \frac{(x - 1) (3 - x)}{x - 1}

= lim_{x \Rightarrow 1} (3 - x) = 3 - 1

=2
Equation 3:

m = lim_{h \Rightarrow 0} \frac{f (a + h) - f (a)}{h}

ii)

m = lim_{h \Rightarrow 0} \frac{f (1 + h) - f (1)}{h}

= lim_{h \Rightarrow 0} \frac{[4 (1 + h) - {(1 + h)}^{2}] - [4 (1) - 1^{2}]}{h}

= lim_{h \Rightarrow 0} \frac{2 h - h^{2}}{h}

= lim_{h \Rightarrow 0} (2 - h) = 2 - 0

=2
Result:
m=2

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