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Questions and Answers
Given $f(x) = \sin(x)$ and $g(x) = 3x$, what is the composite function $f(g(x))$?
Given $f(x) = \sin(x)$ and $g(x) = 3x$, what is the composite function $f(g(x))$?
- $3x + \sin(x)$
- $\sin(x) \cdot 3x$
- $\sin(3x)$ (correct)
- $3\sin(x)$
If $y = \csc^2(2x - \pi)$, which of the following correctly identifies the outer function $f(x)$ and the inner function $g(x)$ in the composition $f(g(x))$?
If $y = \csc^2(2x - \pi)$, which of the following correctly identifies the outer function $f(x)$ and the inner function $g(x)$ in the composition $f(g(x))$?
- $f(x) = x^2$, $g(x) = \csc(2x - \pi)$
- $f(x) = 2x - \pi$, $g(x) = \csc^2(x)$
- $f(x) = \csc^2(x)$, $g(x) = 2x - \pi$ (correct)
- $f(x) = \csc(x)$, $g(x) = 2x - \pi$
If $y = f(u)$ and $u = g(x)$, and both functions are differentiable, the chain rule states:
If $y = f(u)$ and $u = g(x)$, and both functions are differentiable, the chain rule states:
- $\frac{dy}{dx} = \frac{dx}{dy} \cdot \frac{du}{dx}$
- $\frac{dy}{dx} = \frac{dy}{du} + \frac{du}{dx}$
- $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ (correct)
- $\frac{dy}{dx} = \frac{du}{dx} \cdot \frac{dx}{dy}$
To apply the chain rule to a composite function $f(u(x))$, what is the correct order of steps?
To apply the chain rule to a composite function $f(u(x))$, what is the correct order of steps?
Given $y = 3\sqrt{x^3} - 4x$, find its derivative. Assume $f(x) = 3\sqrt{x}$ and $g(x) = x^3 - 4x$.
Given $y = 3\sqrt{x^3} - 4x$, find its derivative. Assume $f(x) = 3\sqrt{x}$ and $g(x) = x^3 - 4x$.
If $y = u^3 - 3u^2 + 1$ and $u = x^2 + 2$, find $\frac{dy}{dx}$.
If $y = u^3 - 3u^2 + 1$ and $u = x^2 + 2$, find $\frac{dy}{dx}$.
Given $f(x) = \sin(3x^2 + 5x - 7)$, find $f'(x)$.
Given $f(x) = \sin(3x^2 + 5x - 7)$, find $f'(x)$.
If $F(x) = \sqrt{x^2 + 1}$, determine $F'(x)$.
If $F(x) = \sqrt{x^2 + 1}$, determine $F'(x)$.
What is the primary goal of using u-substitution in integration?
What is the primary goal of using u-substitution in integration?
What is the first step in applying the substitution rule for integration?
What is the first step in applying the substitution rule for integration?
Evaluate the indefinite integral: $\int (x + 4)^5 dx$.
Evaluate the indefinite integral: $\int (x + 4)^5 dx$.
Evaluate the integral $\int (x^5 + 2)^9 \cdot 5x^4 dx$.
Evaluate the integral $\int (x^5 + 2)^9 \cdot 5x^4 dx$.
Determine the integral $\int \frac{z^2}{\sqrt{1 + z^3}} dz$.
Determine the integral $\int \frac{z^2}{\sqrt{1 + z^3}} dz$.
Compute $\int \frac{x}{\sqrt{x^2 - 1}} dx$.
Compute $\int \frac{x}{\sqrt{x^2 - 1}} dx$.
What is the result of the integral $\int \frac{x^3}{\sqrt{x^3 + 5}} dx$?
What is the result of the integral $\int \frac{x^3}{\sqrt{x^3 + 5}} dx$?
Using the table of integrals, determine the antiderivative of the function $f(x) = \sec(x)$.
Using the table of integrals, determine the antiderivative of the function $f(x) = \sec(x)$.
Using the table of integrals, evaluate $\int \frac{dx}{\sqrt{a^2 - x^2}}$.
Using the table of integrals, evaluate $\int \frac{dx}{\sqrt{a^2 - x^2}}$.
Using the table of integrals, find the integral of $\int \csc(x)\cot(x) dx$
Using the table of integrals, find the integral of $\int \csc(x)\cot(x) dx$
Using the table of integrals, which of the following is the correct formula for $\int \tan(x) dx$?
Using the table of integrals, which of the following is the correct formula for $\int \tan(x) dx$?
According to the provided content, which of the following integrals requires a substitution to solve effectively?
According to the provided content, which of the following integrals requires a substitution to solve effectively?
Given the integral $\int f(g(x)) \cdot g'(x) dx$, what does this expression simplify to after applying u-substitution?
Given the integral $\int f(g(x)) \cdot g'(x) dx$, what does this expression simplify to after applying u-substitution?
Flashcards
What is the Chain Rule?
What is the Chain Rule?
A rule for differentiating composite functions. It states that the derivative of f(g(x)) is f'(g(x)) * g'(x).
What is Composition of Functions?
What is Composition of Functions?
Combining two functions where the output of one function becomes the input of the other, denoted as f(g(x)).
What is u-Substitution?
What is u-Substitution?
A technique used to find the antiderivative of a composite function by reversing the chain rule.
What is an Integral?
What is an Integral?
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Antidifferentiation by Substitution
Antidifferentiation by Substitution
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How do you perform u-substitution?
How do you perform u-substitution?
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In ∫(x + 4)⁵ dx, how do you choose 'u'?
In ∫(x + 4)⁵ dx, how do you choose 'u'?
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How to simplify ∫ (x⁵ + 2)⁹ 5x⁴ dx?
How to simplify ∫ (x⁵ + 2)⁹ 5x⁴ dx?
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How to solve integrals with substitution?
How to solve integrals with substitution?
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What's the final step in u-substitution?
What's the final step in u-substitution?
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Study Notes
- This material covers Lesson 2 of basic calculus, focusing on the fourth quarter.
- It discuses antidifferentiation by substitution and using a table of integrals.
Review Lesson: The Chain Rule for Differentiation
- The chain rule is a method for differentiating composite functions, i.e., functions composed of other functions.
Example 1
- Find f(g(x)) if f(x) = tan(x) and g(x) = 3x + 2
- The result is tan(3x + 2)
Example 2: Decomposition of Functions
- Given a composite function f(g(x)), decomposition involves identifying the outer function f(x) and the inner function g(x).
- y = 3√(x³-4x): f(x) = 3√x, g(x) = x³ - 4x
- y = 1/(2x-3)³: f(x) = 1/x³, g(x) = 2x - 3
- y = sin(3x): f(x) = sin(x), g(x) = 3x
- y = (3x-8)⁴: f(x) = x⁴, g(x) = 3x - 8
- y = csc²(2x - π): f(x) = csc²(x), g(x) = 2x - π
The Chain Rule
- If y = f(u) is a differentiable function of u, and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x.
- The chain rule is expressed as dy/dx = (dy/du) * (du/dx).
- An alternative notation is f'(u) = f'(u) * u'.
- A further notation is d/dx [f(g(x))] = f'(g(x)) * g'(x)
Instructions for the Chain Rule
- To find the derivative f' of a composite function f(u(x)):
- Decompose the function into its outer function f(u) and inner function u(x).
- Differentiate the "mother" function: Find f'(u).
- Differentiate the composed function: Find u'(x).
- Multiply the resultant derivatives: f'(x) = f'(u) * u'(x).
- Substitute for u and simplify the expression.
Chain Rule Derivative Examples
- Find the derivatives of the following functions:
- y = 3√(x³-4x): f(x) = 3√x, g(x) = x³ - 4x
- y = 1/(2x-3)³: f(x) = 1/x³, g(x) = 2x - 3
- y = (3x-8)⁴: f(x) = x⁴, g(x) = 3x - 8
Example 1
- Find dy/dx if y = u³ - 3u² + 1 and u = x² + 2:
- Define f(u) = u³ - 3u² + 1 and u = x² + 2.
- Find the derivatives: f'(u) = 3u² - 6u and u' = 2x.
- Apply the chain rule: dy/dx = f'(u) * u' = (3u² - 6u)(2x).
- Substitute and simplify: dy/dx = 6x⁵ + 12x³.
Example 2
- Differentiate f(x) = sin(3x² + 5x - 7):
- Define f(u) = sin(u) and u = 3x² + 5x - 7.
- Find the derivatives: f'(u) = cos(u) and u' = 6x + 5.
- Apply the chain rule: f'(x) = f'(u) * u' = cos(u)(6x + 5).
- Substitute and simplify: f'(x) = (6x + 5)cos(3x² + 5x - 7).
Example 3
- Find F'(x) if F(x) = √(x² + 1):
- Define f(u) = √u = u^(1/2) and u = x² + 1.
- Find the derivatives: f'(u) = (1/2)u^(-1/2) and u' = 2x.
- Apply the chain rule and simplify: F'(x) = x / √(x² + 1).
Lesson Proper: Antiderivatives via Substitution
- Compute the antiderivative of a function using substitution rule.
- Composite functions are functions combined to make a single one.
- The combination of functions f and g is (f ∘ g)(x) = f(g(x)).
- To apply the Chain Rule on composite functions, find the derivative of the outer function and multiplying it by the derivative of the inner function.
- Symbolically: d/dx [f(g(x))] = f'(g(x)) * g'(x), where f'(g(x)) is the derivative of the outside function and g'(x) is the derivative of the inside function.
- Anti-differentiation by substitution is the inverse of the Chain Rule.
- ∫f(g(x)) * g'(x) dx can be transformed into ∫F(u) + C = ∫f(u) du.
Example 1
- Evaluate ∫(x + 4)⁵ dx:
- Substitute u = x + 4 (du = dx) to simplify the integral to ∫u⁵ du.
- Evaluate ∫u⁵ du as (u⁶)/6 + C.
- Substitute x + 4 back in for u, yielding (x + 4)⁶/6 + C.
Example 2
- Evaluate ∫(x⁵ + 2)⁹ * 5x⁴ dx:
- Substitute u = x⁵ + 2 (du = 5x⁴ dx) to get ∫u⁹ du.
- Evaluate ∫u⁹ du as u¹⁰/10 + C.
- Substitute x⁵ + 2 back in for u, gettting (x⁵ + 2)¹⁰/10 + C.
Example 3
- Evaluate ∫z² / √(1 + z³) dz:
- Substitute u = 1 + z³ (du = 3z² dz, z² dz = (1/3) du) to get (1/3) ∫1/√u du.
- Rewrite as (1/3) ∫u^(-1/2) du, and solve: (2/3)(1 + z³)^(1/2) + C.
Example 4
- Evaluate ∫x / √(x² - 1) dx:
- Substitute u = x² - 1 (du = 2x dx, x dx = (1/2) du). The integral becomes ∫(1/2) / √u du.
- Rewrite as (1/2) ∫u^(-1/2) du and solve to get: (x² - 1)^(1/2) + C.
Example 5
- Evaluate ∫x³ / √(x³ + 5) dx:
- Substitute u = x³ + 5 (x³ = u - 5, du = 3x² dx), transforming the integral.
- Evaluate ∫(u - 5) / √u du to get (2/3)(x³ + 5)^(3/2) - 10(x³ + 5)^(1/2) + C.
Summary Table of Integrals
- Provides a list of common integrals, useful for solving antidifferentiation problems.
- Includes integrals of basic functions like xⁿ, eˣ, aˣ, sin x, cos x, and sec²x, as well as trigonometric functions.
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