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

0. Functions

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

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1:36 minutes

Problem 4.R.118a

Textbook Question

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

a. What is the domain of f (in terms of a)?

Verified step by step guidance

To determine the domain of the function $ f(x) = (a + x)^x $, we need to identify the values of $ x $ for which the expression is defined.

The expression $ (a + x)^x $ is defined when $ a + x > 0 $ because the base of the power must be positive for real number exponents.

Solve the inequality $ a + x > 0 $ to find the values of $ x $. This gives $ x > -a $.

Therefore, the domain of $ f(x) $ in terms of $ a $ is all real numbers $ x $ such that $ x > -a $.

In interval notation, the domain is $ (-a, \infty) $.

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Key Concepts

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = (a + x)ˣ, where a > 0, the domain is determined by the values of x that keep the expression a + x positive, as the base of an exponent must be positive for real-valued outputs.

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = bˣ, where b is a positive constant. In the case of f(x) = (a + x)ˣ, the function exhibits superexponential growth, meaning it grows faster than any polynomial function as x increases. Understanding the behavior of exponential functions is crucial for analyzing their domains and ranges.

Inequalities

Inequalities are mathematical statements that compare two expressions, indicating whether one is greater than, less than, or equal to the other. In determining the domain of f(x) = (a + x)ˣ, we need to solve the inequality a + x > 0, which leads to x > -a. This understanding of inequalities is essential for identifying valid input values for the function.

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Textbook Question

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Find the slope of the secant line that passes through points $A$ and $B$ . Interpret your answer as an average rate of change over the interval $1\leq{t}\leq{3}$ .

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Textbook Question

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Repeat the procedure outlined in part (a) for the secant line that passes through points $P$ and $Q$ .

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Textbook Question

A GPS device tracks the elevation $E$ (in feet) of a hiker walking in the mountains. The elevation $t$ hours after beginning the hike is given in the figure. <IMAGE>

Notice that the curve in the figure is horizontal for an interval of time near $t=5.5$ hr. Give a plausible explanation for the horizontal line segment.

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Textbook Question

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points $P$ and $Q$ on the graph of the function.

a. Sketch a graph of the function and the secant line through $P$ and $Q$ .

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After $t$ seconds, an object dropped from rest falls a distance $d=16t^2$ , where $d$ is measured in feet and $2\leq{t}\leq{5}$ .

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{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).

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Functions and Graphs

Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.

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Horizontal and Vertical Asymptotes

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{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.a. What is the domain of f (in terms of a)?

Key Concepts

Domain of a Function

Exponential Functions

Inequalities

Watch next

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

a. What is the domain of f (in terms of a)?