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hw3 a preview of calculus (target l1, §2.1)
score: 0/4 answered: 0/4
question 1
the point p(3, 17) lies on the curve ( y = x^2 + x + 5 ). if q is the point ( (x, x^2 + x + 5) ), find the slope of the secant line pq for the following values of ( x ).
if ( x = 3.1 ), the slope of pq is:
and if ( x = 3.01 ), the slope of pq is:
and if ( x = 2.9 ), the slope of pq is:
and if ( x = 2.99 ), the slope of pq is:
based on the above results, guess the slope of the tangent line to the curve at ( p(3, 17) ).
question help: video message instructor

Explanation:

The slope of the secant line between two points \( P(x_1,y_1) \) and \( Q(x_2,y_2) \) is given by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( P(3,17) \) and \( Q(x,x^{2}+x + 5) \), so \( y_2=x^{2}+x + 5 \), \( y_1 = 17 \), \( x_2=x \) and \( x_1 = 3 \). So the slope \( m=\frac{(x^{2}+x + 5)-17}{x - 3}=\frac{x^{2}+x-12}{x - 3}\). We can factor the numerator: \( x^{2}+x - 12=(x + 4)(x - 3) \), so the slope simplifies to \( m=x + 4 \) (for \( x
eq3 \)).

Step 1: When \( x = 3.1 \)

We use the formula \( m=x + 4 \). Substitute \( x = 3.1 \) into the formula.
\( m=3.1+4=7.1 \)

Step 2: When \( x = 3.01 \)

Substitute \( x = 3.01 \) into \( m=x + 4 \).
\( m=3.01+4=7.01 \)

Step 3: When \( x = 2.9 \)

Substitute \( x = 2.9 \) into \( m=x + 4 \).
\( m=2.9+4=6.9 \)

Step 4: When \( x = 2.99 \)

Substitute \( x = 2.99 \) into \( m=x + 4 \).
\( m=2.99+4=6.99 \)

Step 5: Guess the slope of the tangent line

As \( x \) approaches \( 3 \) (from both sides, \( x = 3.1,3.01 \) (approaching from the right) and \( x=2.9,2.99 \) (approaching from the left)), the slope of the secant line approaches \( 3 + 4=7 \). So the slope of the tangent line at \( x = 3 \) is \( 7 \).

Answer:

s:

• When \( x = 3.1 \), slope is \( \boldsymbol{7.1} \)
• When \( x = 3.01 \), slope is \( \boldsymbol{7.01} \)
• When \( x = 2.9 \), slope is \( \boldsymbol{6.9} \)
• When \( x = 2.99 \), slope is \( \boldsymbol{6.99} \)
• Slope of tangent line at \( P(3,17) \) is \( \boldsymbol{7} \)