Question

Given a polynomial $P(x) = ax^2 + bx + c$ such that $P''(2)=2$, $P'(2)=0$, and $P(2)=-1$. Find the approximate value of $P(1.001)$ using linear approximation.

• A. 0.002
• B. -0.002
• C. 0.004
• D. -0.004

Hint:

Linear approximation is a powerful tool for estimating function values near a known point. The formula $P(a+h) \approx P(a) + hP'(a)$ is derived from the tangent line equation.

Answer 1

The Correct Option is A

Step 1: Find the first and second derivatives of $P(x)$. \[P(x) = ax^2 + bx + c\] \[P'(x) = 2ax + b\] \[P''(x) = 2a\]
Step 2: Use the given condition $P''(2)=2$ to find the value of $a$. \[P''(2) = 2a\] \[2 = 2a\] \[a = 1\]
Step 3: Use the given condition $P'(2)=0$ and the value of $a$ to find the value of $b$. \[P'(2) = 2a(2) + b\] \[0 = 4a + b\] \[0 = 4(1) + b\] \[b = -4\]
Step 4: Use the given condition $P(2)=-1$ and the values of $a$ and $b$ to find the value of $c$. \[P(2) = a(2)^2 + b(2) + c\] \[-1 = 4a + 2b + c\] \[-1 = 4(1) + 2(-4) + c\] \[-1 = 4 - 8 + c\] \[-1 = -4 + c\] \[c = 3\]
Step 5: Write the complete polynomial $P(x)$ and its derivative $P'(x)$. \[P(x) = x^2 - 4x + 3\] \[P'(x) = 2x - 4\]
Step 6: Apply the linear approximation formula $P(a+h) \approx P(a) + hP'(a)$. Here, we want to approximate $P(1.001)$, so we can set $a=1$ and $h=0.001$. Calculate $P(1)$: \[P(1) = (1)^2 - 4(1) + 3 = 1 - 4 + 3 = 0\] Calculate $P'(1)$: \[P'(1) = 2(1) - 4 = 2 - 4 = -2\]
Step 7: Substitute the values into the linear approximation formula. \[P(1.001) \approx P(1) + (0.001)P'(1)\] \[P(1.001) \approx 0 + (0.001)(-2)\] \[P(1.001) \approx -0.002\]