For a function, f, defined on the set of real numbers, ℝ, it is known that
• the rate of change of f with respect to x is given by $3x^2 - 16x + 11$
• the graph with equation $y = f'(x)$ crosses the x-axis at (7,0) - Scottish Highers Maths - Question 13 - 2019
Question 13
For a function, f, defined on the set of real numbers, ℝ, it is known that
• the rate of change of f with respect to x is given by $3x^2 - 16x + 11$
• the graph with... show full transcript
Worked Solution & Example Answer:For a function, f, defined on the set of real numbers, ℝ, it is known that
• the rate of change of f with respect to x is given by $3x^2 - 16x + 11$
• the graph with equation $y = f'(x)$ crosses the x-axis at (7,0) - Scottish Highers Maths - Question 13 - 2019
Step 1
1. Integrate the Rate of Change
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Answer
To find f(x), we start with the rate of change given by the equation:
f′(x)=3x2−16x+11.
We will integrate this equation with respect to x:
f(x)=∫(3x2−16x+11)dx.
Integrating term by term yields:
f(x)=(33x3−216x2+11x)+c,
which simplifies to:
f(x)=x3−8x2+11x+c.
Step 2
2. Use the Information Given
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Answer
We know that the graph of y=f′(x) crosses the x-axis at (7,0), which means:
f′(7)=0.
Calculating f′(7) using our derivative:
f′(7)=3(7)2−16(7)+11.
Calculating each term gives:
f′(7)=147−112+11=46.
This shows that 7 is not where f′(x) crosses the x-axis. Hence we need to incorporate the condition for f(x) determined by the unknown constant c.
Step 3
3. Solve for c
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Since we are not provided with more context for further values, we use the function f(7):
f(7)=73−8(7)2+11(7)+c.
Calculating each term provides:
f(7)=343−392+77+c=28+c.
So we set f(7) to its known boundary condition (which interacts but doesn't cross):
0=28+c⇒c=−28.
Step 4
4. Final Expression for f(x)
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Answer
Inserting the found value of c into our integrated function, we can express:
f(x)=x3−8x2+11x−28.
This gives us the final expression for f(x) in terms of x.
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