Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 51

Chapter 3, Problem 51

Assume that functions f and g are differentiable with f(1) = 2, f'(1) = −3, g(1) = 4, and g'(1) = −2. Find the equation of the line tangent to the graph of F(x) = f(x)g(x) at x = 1.

Verified step by step guidance

To find the equation of the tangent line to the graph of F(x) = f(x)g(x) at x = 1, we first need to determine the value of F(1). Since F(x) = f(x)g(x), we can substitute x = 1 to get F(1) = f(1)g(1). Given that f(1) = 2 and g(1) = 4, calculate F(1) = 2 * 4.

Next, we need to find the derivative of F(x) = f(x)g(x) using the product rule. The product rule states that if F(x) = f(x)g(x), then F'(x) = f'(x)g(x) + f(x)g'(x).

Substitute the given values into the derivative formula. We have f'(1) = -3, g(1) = 4, f(1) = 2, and g'(1) = -2. Therefore, F'(1) = f'(1)g(1) + f(1)g'(1) = (-3)(4) + (2)(-2).

Now that we have F(1) and F'(1), we can write the equation of the tangent line. The general form of the equation of a tangent line is y = F'(1)(x - 1) + F(1).

Substitute the values of F(1) and F'(1) into the tangent line equation to express the equation of the tangent line at x = 1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product Rule

The product rule is a fundamental differentiation rule used when finding the derivative of a product of two functions. If F(x) = f(x)g(x), then the derivative F'(x) is given by f'(x)g(x) + f(x)g'(x). This rule is essential for determining the slope of the tangent line to the graph of F(x) at a specific point.

Tangent Line Equation

The equation of a tangent line to a curve at a given point is derived using the point-slope form: y - y₁ = m(x - x₁), where m is the slope of the tangent line and (x₁, y₁) is the point of tangency. For F(x) = f(x)g(x), the slope m is F'(1), and the point is (1, F(1)).

Function Evaluation

Function evaluation involves substituting a specific value into a function to find its output. For F(x) = f(x)g(x), evaluating F(1) requires calculating f(1)g(1). This step is crucial for determining the y-coordinate of the point of tangency, which is needed for the tangent line equation.

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Related Practice

Textbook Question

When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on Earth’s surface, depending on the change in g. By keeping track of ΔT, we can estimate the variation in g from the equation T = 2π(L/g)¹/² that relates T, g, and L.

a. With L held constant and g as the independent variable, calculate dT and use it to answer parts (b) and (c).

Textbook Question

Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.

Textbook Question

a. Find an equation for the line that is tangent to the curve y = x³ − 6x² + 5x at the origin.

[Technology Exercise] b. Graph the curve and tangent line together. The tangent line intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

[Technology Exercise] c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent line simultaneously.

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Textbook Question

In Exercises 51 and 52, find dp/dq.

q = (5p² + 2p)⁻³/²

Textbook Question

Find by implicit differentiation.

x²y² = 1

Textbook Question

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

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