Question

the point p(4, 25) lies on the curve y = x² + x + 5. if q is the point (x, x² + x + 5), find the slope of the secant line pq for the following values of x. 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, 25).

Explanation:

Step1: Recall slope - formula

The slope of the secant line $PQ$ where $P(4,25)$ and $Q(x,x^{2}+x + 5)$ is given by $m=\frac{(x^{2}+x + 5)-25}{x - 4}=\frac{x^{2}+x-20}{x - 4}$.

Step2: Factor the numerator

Factor $x^{2}+x - 20=(x + 5)(x - 4)$. So the slope formula becomes $m=\frac{(x + 5)(x - 4)}{x - 4}=x + 5$ for $x
eq4$.

Step3: Calculate slope for $x = 4.1$

Substitute $x = 4.1$ into $m=x + 5$. Then $m=4.1+5=9.1$.

Step4: Calculate slope for $x = 4.01$

Substitute $x = 4.01$ into $m=x + 5$. Then $m=4.01+5=9.01$.

Step5: Calculate slope for $x = 3.99$

Substitute $x = 3.99$ into $m=x + 5$. Then $m=3.99+5=8.99$.

Step6: Calculate slope for $x = 3.9$

Substitute $x = 3.9$ into $m=x + 5$. Then $m=3.9+5=8.9$.

Step7: Guess the slope of the tangent

As $x$ gets closer to $4$, the slope of the secant line approaches the slope of the tangent line. Based on the above - calculated values, the slope of the tangent line at $P(4,25)$ is $9$.

Answer:

When $x = 4.1$, slope of $PQ$ is $9.1$; when $x = 4.01$, slope of $PQ$ is $9.01$; when $x = 3.99$, slope of $PQ$ is $8.99$; when $x = 3.9$, slope of $PQ$ is $8.9$; slope of the tangent line at $P(4,25)$ is $9$.