Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Curve Sketching

Calculus

5. Graphical Applications of Derivatives

Curve Sketching

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1:16 minutes

Problem 3.6.50a

Textbook Question

{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

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.

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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.

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.

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.

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