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enyata [817]
2 years ago
7

Really need to someone to break this down so I can understand it

Mathematics
1 answer:
nignag [31]2 years ago
4 0

Answer:

Part A)

The slope is two.

Part B)

\displaystyle y = 2x - 7

Step-by-step explanation:

Part A)

We want to find the slope of the curve:

\displaystyle y = x^2 - 2x - 3

At the point P(2, -3) by using the limit of the secant slopes through point P.

To find the limit of the secant slopes, we can use the difference quotient. Recall that:

\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Since we want to find the slope of the curve at P(2, -3), <em>x</em> = 2.

Substitute:

\displaystyle f'(2) = \lim_{h \to 0} \frac{f(2 + h) - f(2)}{h}

Simplify. Note that f(2) = -3. Hence:

\displaystyle \begin{aligned}  f'(2) &= \lim_{h\to 0} \frac{\left[(2+h)^2 - 2(2+h) - 3\right] - \left[-3\right]}{h} \\ \\ &=\lim_{h \to 0}\frac{(4 + 4h + h^2)+(-4-2h)+(0)}{h}  \\ \\ &= \lim_{h\to 0} \frac{h^2+2h}{h}\\ \\&=\lim_{h\to 0} h + 2 \\ \\ &= (0) + 2 \\ &= 2\end{aligned}

(Note: I evaluated the limit using direct substitution.)

Hence, the slope of the curve at the point P(2, -3) is two.

Part B)

Since the slope of the curve at point P is two, the slope of the tangent line is also two.

And since we know it passes through the point (2, -3), we can consider using the point-slope form:

\displaystyle y - y_1 = m(x-x_1)

Substitute. <em>m</em> = 2. Therefore, our equation is:

\displaystyle y + 3 = 2(x-2)

We can rewrite this into slope-intercept if desired:

\displaystyle y = 2x - 7

We can verify this by graphing. This is shown below:

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