AP Calculus BC 4.6 Approximating Values of a Function Using Local Linearity and Linearization - MCQs - Exam Style Questions
No-Calc Question
Let \(f\) be a function with \(f(3)=-7\) and \(f'(3)=5\). What is the linear approximation of \(f(2.8)\) using the tangent line at \(x=3\)?
(A) \(-8\)
(B) \(-6\)
(C) \(4\)
(D) \(6\)
(B) \(-6\)
(C) \(4\)
(D) \(6\)
▶️ Answer/Explanation
Detailed solution
Tangent-line (linearization): \(L(x)=f(3)+f'(3)(x-3)=-7+5(x-3)\).
\(L(2.8)=-7+5(-0.2)=-7-1=-8.\)
✅ Answer: (A)
No-Calc Question
Let \(f\) be a differentiable function with \(f(2)=3\) and \(f'(2)=\tfrac12\). Using the line tangent to the graph of \(f\) at \(x=2\) as a local linear approximation for \(f\), what is the estimate for \(f(1.8)\)?
(A) \(2.5\)
(B) \(2.8\)
(C) \(2.9\)
(D) \(3.1\)
(B) \(2.8\)
(C) \(2.9\)
(D) \(3.1\)
▶️ Answer/Explanation
Detailed solution
Tangent (linearization) at \(x=2\):
\(L(x)=f(2)+f'(2)\,(x-2)=3+\tfrac12(x-2)\).
Evaluate at \(x=1.8\):
\(L(1.8)=3+\tfrac12(-0.2)=3-0.1=2.9\).
✅ Answer: (C)