question 3 the point p(25,7) lies on the curve y = √x + 2. if q is the point (x,√x + 2), find the slope of…

question 3 the point p(25,7) lies on the curve y = √x + 2. if q is the point (x,√x + 2), find the slope of the secant line pq for the following values of x. if x = 25.1, the slope of pq is: and if x = 25.01, the slope of pq is: and if x = 24.9, the slope of pq is: and if x = 24.99, the slope of pq is: based on the above results, guess the slope of the tangent line to the curve at p(25,7). question help: video post to forum

question 3 the point p(25,7) lies on the curve y = √x + 2. if q is the point (x,√x + 2), find the slope of the secant line pq for the following values of x. if x = 25.1, the slope of pq is: and if x = 25.01, the slope of pq is: and if x = 24.9, the slope of pq is: and if x = 24.99, the slope of pq is: based on the above results, guess the slope of the tangent line to the curve at p(25,7). question help: video post to forum

Answer

Answer:

  1. When $x = 25.1$:
    • First, find the coordinates of $Q$. When $x = 25.1$, $y=\sqrt{25.1}+2$.
    • The slope formula of the secant - line $PQ$ is $m=\frac{y_Q - y_P}{x_Q - x_P}$. Here, $x_P = 25$, $y_P = 7$, $x_Q = 25.1$, and $y_Q=\sqrt{25.1}+2$.
    • $m_1=\frac{\sqrt{25.1}+2 - 7}{25.1 - 25}=\frac{\sqrt{25.1}-5}{0.1}\approx\frac{5.00999 - 5}{0.1}=\frac{0.00999}{0.1}=0.0999$.
  2. When $x = 25.01$:
    • When $x = 25.01$, $y=\sqrt{25.01}+2$.
    • Using the slope formula $m=\frac{y_Q - y_P}{x_Q - x_P}$, with $x_P = 25$, $y_P = 7$, $x_Q = 25.01$, and $y_Q=\sqrt{25.01}+2$.
    • $m_2=\frac{\sqrt{25.01}+2 - 7}{25.01 - 25}=\frac{\sqrt{25.01}-5}{0.01}\approx\frac{5.000999 - 5}{0.01}=\frac{0.000999}{0.01}=0.0999$.
  3. When $x = 24.9$:
    • When $x = 24.9$, $y=\sqrt{24.9}+2$.
    • Using the slope formula $m=\frac{y_Q - y_P}{x_Q - x_P}$, with $x_P = 25$, $y_P = 7$, $x_Q = 24.9$, and $y_Q=\sqrt{24.9}+2$.
    • $m_3=\frac{\sqrt{24.9}+2 - 7}{24.9 - 25}=\frac{\sqrt{24.9}-5}{-0.1}\approx\frac{4.98999 - 5}{-0.1}=\frac{-0.01001}{-0.1}=0.1001$.
  4. When $x = 24.99$:
    • When $x = 24.99$, $y=\sqrt{24.99}+2$.
    • Using the slope formula $m=\frac{y_Q - y_P}{x_Q - x_P}$, with $x_P = 25$, $y_P = 7$, $x_Q = 24.99$, and $y_Q=\sqrt{24.99}+2$.
    • $m_4=\frac{\sqrt{24.99}+2 - 7}{24.99 - 25}=\frac{\sqrt{24.99}-5}{-0.01}\approx\frac{4.999001 - 5}{-0.01}=\frac{-0.000999}{-0.01}=0.0999$.
  5. Guess the slope of the tangent - line:
    • Based on the above results, as $x$ gets closer to $25$, the slope of the secant - line $PQ$ approaches $0.1$.

So:

  • When $x = 25.1$, the slope of $PQ$ is $0.0999$.
  • When $x = 25.01$, the slope of $PQ$ is $0.0999$.
  • When $x = 24.9$, the slope of $PQ$ is $0.1001$.
  • When $x = 24.99$, the slope of $PQ$ is $0.0999$.
  • The guessed slope of the tangent line at $P(25,7)$ is $0.1$.

Explanation:

Step1: Recall slope formula

$m=\frac{y_2 - y_1}{x_2 - x_1}$

Step2: For $x = 25.1$

Find $y_Q=\sqrt{25.1}+2$, then $m_1=\frac{\sqrt{25.1}+2 - 7}{25.1 - 25}$

Step3: For $x = 25.01$

Find $y_Q=\sqrt{25.01}+2$, then $m_2=\frac{\sqrt{25.01}+2 - 7}{25.01 - 25}$

Step4: For $x = 24.9$

Find $y_Q=\sqrt{24.9}+2$, then $m_3=\frac{\sqrt{24.9}+2 - 7}{24.9 - 25}$

Step5: For $x = 24.99$

Find $y_Q=\sqrt{24.99}+2$, then $m_4=\frac{\sqrt{24.99}+2 - 7}{24.99 - 25}$

Step6: Guess tangent - slope

Observe the trend of secant - slopes as $x\to25$.