Question

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

Answer 1

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), x = 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. m = 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: