Question

let ( f(x) = x^3 - 15x ). calculate the difference quotient ( \frac{f(3+h) - f(3)}{h} ) for ( h = .1 ), ( h = .01 ), ( h = -.01 ), ( h = -1 ). if someone now told you that the derivative (slope of the tangent line to the graph) of ( f(x) ) at ( x = 3 ) was an integer, what would you expect it to be? note: you can earn partial credit on this problem. preview my answers submit answers your score was recorded. your score was successfully sent to canvas. you have attempted this problem 2 times. you received a score of 0% for this attempt. your overall recorded score is 0%. you have unlimited attempts remaining. email instructor

Explanation:

Step 1: Find \( f(3) \)

Given \( f(x) = x^3 - 15x \), substitute \( x = 3 \):
\( f(3) = 3^3 - 15\times3 = 27 - 45 = -18 \)

Step 2: Find \( f(3 + h) \)

Substitute \( x = 3 + h \) into \( f(x) \):
\( f(3 + h) = (3 + h)^3 - 15(3 + h) \)
Expand \( (3 + h)^3 \): \( 27 + 27h + 9h^2 + h^3 \)
Expand \( -15(3 + h) \): \( -45 - 15h \)
Combine terms: \( f(3 + h) = 27 + 27h + 9h^2 + h^3 - 45 - 15h = h^3 + 9h^2 + 12h - 18 \)

Step 3: Compute the difference quotient \( \frac{f(3 + h) - f(3)}{h} \)

Substitute \( f(3 + h) \) and \( f(3) \):
\( \frac{(h^3 + 9h^2 + 12h - 18) - (-18)}{h} = \frac{h^3 + 9h^2 + 12h}{h} \)
Factor out \( h \) in the numerator: \( \frac{h(h^2 + 9h + 12)}{h} \)
Cancel \( h \) (for \( h
eq 0 \)): \( h^2 + 9h + 12 \)

Step 4: Evaluate for different \( h \) values

• For \( h = 0.1 \):

\( (0.1)^2 + 9(0.1) + 12 = 0.01 + 0.9 + 12 = 12.91 \)

• For \( h = 0.01 \):

\( (0.01)^2 + 9(0.01) + 12 = 0.0001 + 0.09 + 12 = 12.0901 \)

• For \( h = -0.01 \):

\( (-0.01)^2 + 9(-0.01) + 12 = 0.0001 - 0.09 + 12 = 11.9101 \)

• For \( h = -0.1 \):

\( (-0.1)^2 + 9(-0.1) + 12 = 0.01 - 0.9 + 12 = 11.11 \)

Step 5: Find the derivative (slope of tangent) at \( x = 3 \)

The derivative \( f'(x) = 3x^2 - 15 \). Substitute \( x = 3 \):
\( f'(3) = 3(3)^2 - 15 = 27 - 15 = 12 \)

Answer:

s:

• For \( h = 0.1 \): \( 12.91 \)
• For \( h = 0.01 \): \( 12.0901 \)
• For \( h = -0.01 \): \( 11.9101 \)
• For \( h = -0.1 \): \( 11.11 \)
• Derivative (slope) at \( x = 3 \): \( 12 \)