QQuestionMathematics
QuestionMathematics
Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = - 3 and g'(5) = 6
Enter your answer in terms of y and x.
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Answer
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Step 1:: Recall the point-slope form of a linear equation, which is y - y\_1 = m(2$).
Here, m is the slope and (x\_1, y\_1) is a point on the line. In this case, we are given the slope g'(5) = 6 and a point (5, g(2$)) = (5, - 3). We will use this information to find the equation of the tangent line.
Step 2:: Since the slope of the tangent line is given as g'(5) = 6, we can directly use this value for m in the point-slope form.
Thus, the equation of the tangent line becomes: y - y\_1 = 6(x - x\_1)
Step 3:: Now, we substitute the point (5, - 3) into the equation:
y - (- 3) = 6(x - 5)
Step 4:: Simplify the equation:
y = 6x - 30 - 3
Step 5:: Further simplification:
y = 6x - 33
Final Answer
The equation of the tangent line to the graph of y = g(x) at x = 5 is y = 6x - 33.
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