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

7. Antiderivatives & Indefinite Integrals

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

Previous problemNext problem

1:34 minutes

Problem 4.9.9

Textbook Question

If F(x) = x² - 3x + C and F (-1) = 4 , what is the value of C?

Verified step by step guidance

Start with the given function F(x) = x² - 3x + C, where C is a constant to be determined.

Substitute x = -1 into the function F(x) since it is given that F(-1) = 4. This gives the equation: F(-1) = (-1)² - 3(-1) + C.

Simplify the terms in the equation: (-1)² = 1, -3(-1) = 3, so the equation becomes F(-1) = 1 + 3 + C.

Set F(-1) equal to 4 as given in the problem: 4 = 1 + 3 + C.

Solve for C by isolating it: Subtract 1 and 3 from both sides of the equation to find the value of C.

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Key Concepts

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

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this case, we need to evaluate the function F(x) at x = -1 to find the corresponding output, which is given as 4. This process is fundamental in calculus as it allows us to analyze the behavior of functions at specific points.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange and simplify algebraic expressions. To find the value of C in the equation F(-1) = 4, we will substitute -1 into the function F(x) and solve for C. Mastery of algebraic manipulation is essential for solving equations and understanding calculus concepts.

Constant Term in Functions

In the context of functions, a constant term is a fixed value that does not change with the input variable. In the function F(x) = x² - 3x + C, the term C represents a constant that shifts the graph of the function vertically. Understanding how constant terms affect function values is crucial for analyzing and interpreting functions in calculus.