Question
1. If an equation of the tangent line to the curve \( y=f(x) \) at the point where \( a=2 \) is \( y=4 x-5 \), find \( f(2) \) and \( f^{\prime}(2) \).
Ask by Hanson Washington. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(2) = 3 \) and \( f^{\prime}(2) = 4 \).
Solution
To determine \( f(2) \) and \( f^{\prime}(2) \) from the equation of the tangent line \( y = 4x - 5 \) to the curve \( y = f(x) \) at \( x = 2 \), we can follow these steps:
1. **Equation of the Tangent Line:**
The general form of the tangent line to \( y = f(x) \) at \( x = a \) is:
\[
y = f(a) + f^{\prime}(a)(x - a)
\]
Given \( a = 2 \), the tangent line equation becomes:
\[
y = f(2) + f^{\prime}(2)(x - 2)
\]
2. **Compare with the Given Tangent Line:**
The given tangent line equation is:
\[
y = 4x - 5
\]
By comparing the two equations:
\[
f(2) + f^{\prime}(2)(x - 2) = 4x - 5
\]
3. **Determine \( f^{\prime}(2) \):**
Compare the coefficients of \( x \):
\[
f^{\prime}(2) = 4
\]
4. **Determine \( f(2) \):**
Compare the constant terms:
\[
f(2) - 2f^{\prime}(2) = -5
\]
Substitute \( f^{\prime}(2) = 4 \):
\[
f(2) - 2(4) = -5 \implies f(2) = -5 + 8 \implies f(2) = 3
\]
**Final Answer:**
\[
f(2) = 3 \quad \text{and} \quad f^{\prime}(2) = 4
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To find \( f(2) \) and \( f^{\prime}(2) \) from the equation of the tangent line, we can interpret the equation \( y = 4x - 5 \). The slope of the tangent line, \( f^{\prime}(2) \), is given by the coefficient of \( x \), which is \( 4 \). Therefore, \( f^{\prime}(2) = 4 \). Next, we need to find \( f(2) \). We can find the specific point on the tangent line where \( x = 2 \): \[ y = 4(2) - 5 = 8 - 5 = 3. \] Thus, at \( x = 2 \), \( f(2) = 3 \). To summarize: - \( f(2) = 3 \) - \( f^{\prime}(2) = 4 \)
