Skip to main content
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 3.6.50a
Chapter 3, Problem 3.6.50a

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².
a. Graph the height function.

Verified step by step guidance
1
Understand the function: The height of the tree, h, is given by the quadratic function h = 5.67 + 0.70b + 0.0067b², where b is the base diameter in centimeters and h is the height in meters.
Identify the type of function: This is a quadratic function in terms of b, which means its graph will be a parabola. Since the coefficient of b² is positive (0.0067), the parabola opens upwards.
Determine the vertex: The vertex of a parabola given by the function h = ax² + bx + c can be found using the formula b = -B/(2A), where A is the coefficient of b² and B is the coefficient of b. In this case, A = 0.0067 and B = 0.70.
Calculate the vertex: Substitute A and B into the vertex formula to find the base diameter b at the vertex. Then, substitute this value of b back into the original function to find the corresponding height h.
Plot the graph: Use the vertex and additional points by choosing various values of b to calculate corresponding h values. Plot these points on a graph with b on the x-axis and h on the y-axis to visualize the height function as a parabola opening upwards.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Quadratic Functions

The given height function h = 5.67 + 0.70b + 0.0067b² is a quadratic function, which is characterized by its parabolic shape. Quadratic functions can be represented in the standard form ax² + bx + c, where a, b, and c are constants. The coefficient of the b² term (0.0067) indicates that the parabola opens upwards, and its vertex represents the minimum point of the function.
Recommended video:
6:04
Introduction to Polynomial Functions

Graphing Functions

Graphing a function involves plotting points on a coordinate system to visualize the relationship between the variables. For the height function, the x-axis can represent the base diameter (b), while the y-axis represents the height (h). Understanding how to identify key features such as intercepts, vertex, and the direction of the parabola is essential for accurately graphing the function.
Recommended video:
5:53
Graph of Sine and Cosine Function

Interpreting Parameters

In the function h = 5.67 + 0.70b + 0.0067b², the parameters have specific meanings: the constant term (5.67) represents the height when the base diameter is zero, the linear term (0.70b) indicates the rate of change of height with respect to diameter, and the quadratic term (0.0067b²) shows how the growth rate changes as the diameter increases. Understanding these parameters helps in interpreting the function's behavior and its implications for tree growth.
Recommended video:
03:55
The Quotient Rule Example 5
Related Practice
Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

³√x+³√y⁴ = 2;(1,1)

194
views
Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(s) = 4s³+3s; a= -3, -1

233
views
Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

286
views
Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √3x; a= 12

184
views
Textbook Question

Deriving trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

413
views
Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

a. Use implicit differentiation to find dy/dx.

282
views