Question

the point (p(4,6)) lies on the curve (y = sqrt{x}+4). if (q) is the point ((x,sqrt{x}+4)), find the slope of the secant line (pq) for the following values of (x). enter your answer as a decimal, be sure to round correctly. if (x = 4.1), the slope of (pq) is: and if (x = 4.01), the slope of (pq) is: and if (x = 3.9), the slope of (pq) is: and if (x = 3.99), the slope of (pq) is: based on the above results, guess the slope of the tangent line to the curve at (p(4,6)). question help: video message instructor submit question

Explanation:

Step1: Recall slope formula

The slope of the secant line between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $P(4,6)$ so $x_1 = 4,y_1=6$ and $Q(x,\sqrt{x}+4)$ so $x_2=x,y_2=\sqrt{x}+4$. The slope formula becomes $m=\frac{\sqrt{x}+4 - 6}{x - 4}=\frac{\sqrt{x}-2}{x - 4}$.

Step2: When $x = 4.1$

Substitute $x = 4.1$ into the slope formula:
\[

$$\begin{align*} m&=\frac{\sqrt{4.1}-2}{4.1 - 4}\\ &=\frac{\sqrt{4.1}-2}{0.1}\\ &\approx\frac{2.0248 - 2}{0.1}\\ &=\frac{0.0248}{0.1}\\ &= 0.248 \end{align*}$$

\]

Step3: When $x = 4.01$

Substitute $x = 4.01$ into the slope formula:
\[

$$\begin{align*} m&=\frac{\sqrt{4.01}-2}{4.01 - 4}\\ &=\frac{\sqrt{4.01}-2}{0.01}\\ &\approx\frac{2.0025 - 2}{0.01}\\ &=\frac{0.0025}{0.01}\\ &=0.25 \end{align*}$$

\]

Step4: When $x = 3.9$

Substitute $x = 3.9$ into the slope formula:
\[

$$\begin{align*} m&=\frac{\sqrt{3.9}-2}{3.9 - 4}\\ &=\frac{\sqrt{3.9}-2}{- 0.1}\\ &\approx\frac{1.9748 - 2}{-0.1}\\ &=\frac{-0.0252}{-0.1}\\ &=0.252 \end{align*}$$

\]

Step5: When $x = 3.99$

Substitute $x = 3.99$ into the slope formula:
\[

$$\begin{align*} m&=\frac{\sqrt{3.99}-2}{3.99 - 4}\\ &=\frac{\sqrt{3.99}-2}{-0.01}\\ &\approx\frac{1.9975 - 2}{-0.01}\\ &=\frac{-0.0025}{-0.01}\\ &= 0.25 \end{align*}$$

\]

Step6: Guess the slope of the tangent line

As $x$ gets closer to $4$ from both sides, the slopes of the secant lines are approaching $0.25$. So we guess the slope of the tangent line at $P(4,6)$ is $0.25$.

Answer:

When $x = 4.1$, the slope is $0.248$.
When $x = 4.01$, the slope is $0.25$.
When $x = 3.9$, the slope is $0.252$.
When $x = 3.99$, the slope is $0.25$.
The guessed slope of the tangent line at $P(4,6)$ is $0.25$.