Question
Suppose an equation of the tangent line to the curve \( y=f(x) \) at the point where \( a=7 \) is \( y=4 x-5 \). Find \( f(7) \). \( f(7)= \) Find \( f^{\prime}(7) \). \( f^{\prime}(7)= \)
Ask by Tucker Hamilton. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(7) = 23 \) and \( f^{\prime}(7) = 4 \).
Solution
To determine \( f(7) \) and \( f^{\prime}(7) \), we can use the information provided by the tangent line equation.
**Given:**
- The tangent line to the curve \( y = f(x) \) at \( x = 7 \) is \( y = 4x - 5 \).
**1. Finding \( f(7) \):**
The tangent line passes through the point \( (7, f(7)) \). Substitute \( x = 7 \) into the tangent line equation to find \( f(7) \):
\[
f(7) = 4(7) - 5 = 28 - 5 = 23
\]
**2. Finding \( f^{\prime}(7) \):**
The slope of the tangent line represents the derivative of \( f(x) \) at \( x = 7 \). From the tangent line equation \( y = 4x - 5 \), the slope is 4. Therefore:
\[
f^{\prime}(7) = 4
\]
**Final Answers:**
\[
f(7) = 23
\]
\[
f^{\prime}(7) = 4
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find \( f(7) \), we observe that the equation of the tangent line at the point \( a = 7 \) is given as \( y = 4x - 5 \). The value of \( f(7) \) corresponds to the \( y \)-coordinate of the point where the tangent line touches the curve \( y = f(x) \). Substituting \( x = 7 \) into the tangent line equation gives: \[ y = 4(7) - 5 = 28 - 5 = 23 \] Thus, \( f(7) = 23 \). Next, to find \( f^{\prime}(7) \), we recognize that the slope of the tangent line represents the derivative at that point. From the tangent line's equation \( y = 4x - 5 \), we see that the slope is 4. Therefore, \[ f^{\prime}(7) = 4. \] So, in summary: \( f(7) = 23 \) and \( f^{\prime}(7) = 4 \).
