Let $f$ be a function such that $f^{\prime}(x)=2 \cos (2 x)$ and $f\left(\frac{2 \pi}{3}\right)=-\frac{\sqrt{3}}{2}$. Use the tangent line approximation of $f(x)$ at $x=\frac{2 \pi}{3}$ to estimate the value of $f(\pi)$.
a $-\frac{\sqrt{3}}{2}-\frac{\pi}{3}$
b $-\frac{5 \pi}{3}+\frac{1}{2}$
c $\frac{2 \pi}{3}-\frac{2}{\sqrt{3}}$
d $-\frac{\sqrt{3}}{3}$
e $\frac{2 \pi}{3}$
Answer (1)
Find the slope of the tangent line at x = 3 2 π by evaluating f ′ ( 3 2 π ) = − 1 .
Write the equation of the tangent line: y = − 1 ( x − 3 2 π ) − 2 3 .
Substitute x = π into the tangent line equation: f ( π ) ≈ − 3 π − 2 3 .
The estimated value of f ( π ) is − 2 3 − 3 π .
Explanation
Problem Setup We are given the derivative of a function f ′ ( x ) = 2 cos ( 2 x ) and a point on the function f ( 3 2 π ) = − 2 3 . We want to estimate the value of f ( π ) using the tangent line approximation at x = 3 2 π .
Finding the Slope First, we need to find the slope of the tangent line at x = 3 2 π . This is given by evaluating the derivative at this point: f ′ ( 3 2 π ) = 2 cos ( 2 ⋅ 3 2 π ) = 2 cos ( 3 4 π ) . Since cos ( 3 4 π ) = − 2 1 , we have f ′ ( 3 2 π ) = 2 ⋅ ( − 2 1 ) = − 1 .
Equation of the Tangent Line Now we can write the equation of the tangent line at x = 3 2 π . The tangent line is given by y = f ′ ( x 0 ) ( x − x 0 ) + f ( x 0 ) , where x 0 = 3 2 π . So, the equation of the tangent line is y = − 1 ( x − 3 2 π ) − 2 3 .
Estimating f(pi) To estimate f ( π ) , we substitute x = π into the tangent line equation: f ( π ) ≈ − 1 ( π − 3 2 π ) − 2 3 = − 1 ( 3 π ) − 2 3 = − 3 π − 2 3 .
Final Answer Therefore, the estimated value of f ( π ) is − 3 π − 2 3 . Comparing this to the answer choices, we see that it matches option a.
Conclusion The final answer is − 2 3 − 3 π .
Examples
Tangent line approximations are used in various fields, such as physics and engineering, to estimate the behavior of complex functions near a specific point. For example, when analyzing the motion of a pendulum, the small-angle approximation uses a tangent line to simplify the equations of motion, making them easier to solve. This allows engineers to quickly estimate the pendulum's period and behavior without needing to solve more complex, non-linear equations. Similarly, in economics, tangent lines can approximate cost or revenue functions to make quick predictions about marginal cost or revenue.
