Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.1.5

Chapter 3, Problem 3.1.5

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = 4 − x², (−1, 3)

Verified step by step guidance

First, identify the function given: \( y = 4 - x^2 \). This is a quadratic function representing a parabola that opens downwards.

To find the equation of the tangent line, we need the derivative of the function, which gives us the slope of the tangent line at any point \( x \). Differentiate \( y = 4 - x^2 \) with respect to \( x \) to get \( \frac{dy}{dx} = -2x \).

Evaluate the derivative at the given point \( x = -1 \) to find the slope of the tangent line. Substitute \( x = -1 \) into \( \frac{dy}{dx} = -2x \) to get the slope \( m = -2(-1) = 2 \).

Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (-1, 3) \) and \( m = 2 \). Substitute these values into the equation to get \( y - 3 = 2(x + 1) \).

Simplify the equation \( y - 3 = 2(x + 1) \) to get the final equation of the tangent line. This will give you the equation in the form \( y = mx + b \), which can be used to sketch the tangent line along with the curve \( y = 4 - x^2 \).

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Key Concepts

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

Derivative

The derivative of a function at a point provides the slope of the tangent line to the curve at that point. For the function y = 4 - x², the derivative is found using basic differentiation rules, resulting in dy/dx = -2x. Evaluating this derivative at x = -1 gives the slope of the tangent line at the point (-1, 3).

Point-Slope Form of a Line

The point-slope form is a method for writing the equation of a line when you know a point on the line and its slope. It is expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point. Using the slope from the derivative and the point (-1, 3), you can find the equation of the tangent line.

Graphing Functions and Tangent Lines

Graphing involves plotting the curve of the function and the tangent line to visualize their relationship. For y = 4 - x², the graph is a downward-opening parabola. The tangent line at (-1, 3) will touch the curve at this point, illustrating the concept of tangency where the line just 'kisses' the curve without crossing it.

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Slopes of Tangent Lines

Related Practice

Textbook Question

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder’s volume is V = πh³. The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h.

Textbook Question

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

f(x) = √(x + 1), (8, 3)

Textbook Question

Tangent line to y = √x Does any tangent line to the curve y = √x cross the x-axis at x = −1? If so, find an equation for the line and the point of tangency. If not, why not?

Textbook Question

Derivatives

In Exercises 1–18, find dy/dx.

f(x) = x³ sin x cos x

Textbook Question

Derivatives

In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)

Textbook Question

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1