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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 74b
Chapter 3, Problem 74b

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.
b. Find an equation of the line tangent to y = h(x) at x=2.

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1
Step 1: Identify the function h(x) = \(\frac{f(x)}{x - 3}\) and the point of tangency x = 2.
Step 2: Calculate h(2) using the given f(2) = 2. Substitute x = 2 into h(x) to find h(2).
Step 3: Use the quotient rule to find h'(x). The quotient rule states that if h(x) = \(\frac{u(x)}{v(x)}\), then h'(x) = \(\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, u(x) = f(x) and v(x) = x - 3.
Step 4: Calculate h'(2) using the values f(2) = 2, f'(2) = 3, and v(2) = 2 - 3. Substitute these into the derivative found in Step 3.
Step 5: Use the point-slope form of a line, y - y_1 = m(x - x_1), where m = h'(2) and the point (x_1, y_1) is (2, h(2)), to write the equation of the tangent line.

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Key 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 is given by the derivative of the function at that point. In this case, to find the equation of the tangent line to y = h(x) at x = 2, we need to evaluate h(2) and h'(2).
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Slopes of Tangent Lines

Derivative

The derivative of a function measures 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. For the function h(x), we need to apply the quotient rule to find h'(x), which will help us determine the slope of the tangent line at x = 2.
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Derivatives

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If h(x) = f(x) / g(x), the derivative h'(x) is given by (g(x)f'(x) - f(x)g'(x)) / (g(x))². In this problem, we will apply the quotient rule to differentiate h(x) = f(x) / (x - 3) to find the necessary slope for the tangent line.
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The Quotient Rule
Related Practice
Textbook Question

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).

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Textbook Question

{Use of Tech} The Witch of Agnesi The graph of y = a3 / (x2 + a2), where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

Let a = 3 and find an equation of the line tangent to y = 27 / (x2 + 9) at x = 2.

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Textbook Question

Find an equation of the line tangent to the following curves at the given value of x.

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Textbook Question

Calculate the derivative of the following functions.

y = (p+3)² sin p²

Textbook Question

27–76. Calculate the derivative of the following functions.

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Textbook Question

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = g(x) at x = 3.

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