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Questions and Answers
What is the average rate of change of the function $f(x) = 2 - 2x + 3$ as $x$ changes from 3 to 11?
Using the definition of the derivative, what is the derivative of $f(x) = 2 - 2x + 3$ at $x = 3$?
What is the speed of the particle at time $t = 1$ for the position function $p(t) = igg( rac{1}{t + 1}, 1 + e^{2t}igg)$?
For the function $f(x) = x^4 - 18x^2 - 1$, what intervals is $f(x)$ concave up?
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Which of the following correctly identifies the local maximum points of the function $f(x) = x^4 - 18x^2 - 1$?
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What is the equation of the tangent line to the function $f(x) = x^3 - 2x^2 + x$ at the point where $x = 2$?
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What is the value of $f^{-1}(1)$ for the function $f(x) = rac{1}{x+1} + ext{sin}(x)$?
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What is the derivative $(f^{-1})'(1)$ for the function $f(x) = rac{1}{x+1} + ext{sin}(x)$ at the point where $f(x) = 1$?
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Study Notes
Average Rate of Change
- The average rate of change of a function f(x) over an interval from x = a to x = b is calculated as: (f(b) - f(a))/(b - a)
- In this case, f(x) = √(2 - 2x + 3) and the interval is from x = 3 to x = 11
Derivative Definition
- The derivative of a function f(x) at a point x = a, denoted as f'(a), is defined as the limit of the difference quotient as h approaches 0: f'(a) = lim(h->0) [(f(a + h) - f(a))/h]
Particle Motion
- The position of a particle at time t is given by p(t) = (1/(t + 1), 1 + e^(2t))
- The speed of the particle at time t is the magnitude of the velocity vector, which is the derivative of the position vector: |p'(t)|
- The acceleration of the particle at time t is the derivative of the velocity vector: p''(t)
Concavity and Critical Points
- A function is concave up on an interval where its second derivative is positive, and concave down where its second derivative is negative.
- Critical points of a function are points where the derivative is either zero or undefined.
- A function is monotone increasing on an interval where its derivative is positive, and monotone decreasing where its derivative is negative.
- Local minimum points and local maximum points occur at critical points where the derivative changes sign.
Differentiation Rules
- The derivative of f(x) = (cos x)e^(x+1) + 2sin x is f'(x) = -sin(x)e^(x+1) + (cos x)e^(x+1) + 2cos(x)
- The derivative of f(x) = e^(ln(2x - x^3)) + (cos(3x))/(x^2 + 4) is f'(x) = (2 - 3x^2)e^(ln(2x - x^3)) - (3sin(3x)(x^2 + 4) - 2xcos(3x))/(x^2 + 4)^2
- The derivative of f(x) = (cos x)/(x^2 + 4) is f'(x) = (-sin(x)(x^2 + 4) - 2xcos(x))/(x^2 + 4)^2
Logarithmic Differentiation
- Logarithmic differentiation is a technique used to find the derivative of functions that are products, quotients, or powers of functions
- To use logarithmic differentiation, take the natural logarithm of both sides of the equation, simplify, and then differentiate implicitly
Tangent Line
- The equation of the tangent line to a curve at a point (a, f(a)) is given by y - f(a) = f'(a)(x - a)
Inverse Functions
- An inverse function reverses the input and output of the original function.
- The derivative of an inverse function at a point f(a) is given by (f^-1)'(f(a)) = 1/f'(a)
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Description
This quiz covers key concepts in calculus including the average rate of change, derivatives, and particle motion. You'll explore the definitions and calculations related to functions, as well as concavity and critical points. Test your understanding of these fundamental topics in calculus to enhance your learning experience.
