Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 46

Chapter 3, Problem 46

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.

Verified step by step guidance

First, find the derivative of each curve to determine the slope of the tangent line. For the curve y = x² + ax + b, the derivative is dy/dx = 2x + a. For the curve y = cx - x², the derivative is dy/dx = c - 2x.

Since the curves have a common tangent line at the point (1,0), the slopes of the tangent lines at x = 1 must be equal. Set the derivatives equal at x = 1: 2(1) + a = c - 2(1).

Solve the equation from the previous step to find a relationship between a and c: 2 + a = c - 2.

Next, ensure that the point (1,0) lies on both curves. Substitute x = 1 and y = 0 into each curve equation. For y = x² + ax + b, substitute to get 0 = 1² + a(1) + b. For y = cx - x², substitute to get 0 = c(1) - 1².

Solve the system of equations obtained from the previous steps to find the values of a, b, and c. Use the relationships: 0 = 1 + a + b and 0 = c - 1, along with the equation 2 + a = c - 2, to find the values of a, b, and c.

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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 and has the same slope as the curve at that point. To find the tangent line, we need to calculate the derivative of the function at the point of tangency, which gives us the slope, and then use the point-slope form of a line to express the tangent.

Derivatives

Derivatives represent the rate of change of a function with respect to its variable. In this context, we will differentiate both quadratic functions to find their slopes at the point (1,0). Setting these derivatives equal at the point of tangency will help us establish a relationship between the coefficients a, b, and c.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax² + bx + c. In this problem, we are dealing with two specific quadratic functions, and understanding their properties, such as vertex, axis of symmetry, and how they can intersect or share tangent lines, is crucial for solving for the unknown coefficients.

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Related Practice

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

Slopes and Tangent Lines

a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.

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Textbook Question

Find by implicit differentiation.

x²y² = 1

Textbook Question

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.

Textbook Question

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

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Textbook Question

Find by implicit differentiation.

x² + xy + y² - 5x = 2