Textbook Question
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = √3x; a= 12
2. Intro to Derivatives
Tangent Lines and Derivatives
2:56 minutesProblem 3.1.22a
Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = -7x; P(-1,7)
Verified step by step guidance1
Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function \( f(x) \) at a point \( x = a \) is given by \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
Step 2: Identify the function \( f(x) = -7x \) and the point \( P(-1, 7) \). Here, \( a = -1 \) and \( f(a) = 7 \).
Step 3: Substitute \( f(x) = -7x \) into the derivative definition: \( f'(a) = \lim_{h \to 0} \frac{f(-1+h) - f(-1)}{h} \).
Step 4: Calculate \( f(-1+h) \) and \( f(-1) \). Since \( f(x) = -7x \), we have \( f(-1+h) = -7(-1+h) = 7 - 7h \) and \( f(-1) = -7(-1) = 7 \).
Step 5: Substitute these values into the limit expression: \( f'(a) = \lim_{h \to 0} \frac{(7 - 7h) - 7}{h} \). Simplify the expression and evaluate the limit to find the slope of the tangent line.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.
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Slopes of Tangent Lines
Derivative
The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, finding the derivative of f(x) = -7x will provide the slope of the tangent line at point P.
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Point-Slope Form
The point-slope form of a linear equation is used to express the equation of a line given a point on the line and its slope. It is written as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is useful for constructing the equation of the tangent line once the slope is determined.
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3:56Slope-Intercept Form
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