Podcast
Questions and Answers
What does the average rate of change represent graphically?
What does the average rate of change represent graphically?
- The area under the curve
- The slope of the tangent line
- The slope of the secant line (correct)
- The second derivative of the function
According to the power rule, what is the derivative of $f(x) = x^n$?
According to the power rule, what is the derivative of $f(x) = x^n$?
- $f'(x) = x^n$
- $f'(x) = nx^{n-1}$ (correct)
- $f'(x) = nx^{n+1}$
- $f'(x) = x^{n-1}$
If $y = f(u)$ and $u = g(x)$, according to the chain rule (Leibniz form), what is $\frac{dy}{dx}$?
If $y = f(u)$ and $u = g(x)$, according to the chain rule (Leibniz form), what is $\frac{dy}{dx}$?
- $\frac{dy}{dx} = \frac{du}{dx}$
- $\frac{dy}{dx} = \frac{dy}{dx} \cdot \frac{du}{dy}$
- $\frac{dy}{dx} = \frac{dx}{du} \cdot \frac{dy}{du}$
- $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ (correct)
If $y = c \cdot f(x)$, where c is a constant, what is $y'$?
If $y = c \cdot f(x)$, where c is a constant, what is $y'$?
What is the derivative of a constant function $f(x) = c$?
What is the derivative of a constant function $f(x) = c$?
Given $y = f \cdot g$, what is $y'$ according to the product rule?
Given $y = f \cdot g$, what is $y'$ according to the product rule?
If $y = \frac{f}{g}$, what is $y'$?
If $y = \frac{f}{g}$, what is $y'$?
If $y = f + g$, what is $y'$?
If $y = f + g$, what is $y'$?
Given the function $f(x) = 4x^3 - 5x$, what is the average rate of change from $x = 0$ to $x = 2$?
Given the function $f(x) = 4x^3 - 5x$, what is the average rate of change from $x = 0$ to $x = 2$?
What is the instantaneous rate of change of a function at a point equivalent to?
What is the instantaneous rate of change of a function at a point equivalent to?
Find dy/dx if $y = x^7 + 3x^2 - 5x + 6 $
Find dy/dx if $y = x^7 + 3x^2 - 5x + 6 $
What is the equivalent of $x^{-n}$?
What is the equivalent of $x^{-n}$?
Determine the average rate of change of $y = x(x-1)^2$ from $x = -1$ to $x = 2$.
Determine the average rate of change of $y = x(x-1)^2$ from $x = -1$ to $x = 2$.
What is the slope of the tangent line for the function $f(x) = 4x^2 - 7$ at the point (3,5)?
What is the slope of the tangent line for the function $f(x) = 4x^2 - 7$ at the point (3,5)?
What is the slope of the tangent line for the function $y = -2x^2(x - 3)$ at $x = 1$?
What is the slope of the tangent line for the function $y = -2x^2(x - 3)$ at $x = 1$?
If a ball's height is given by $h(t) = -12t^2 + 75t$, what is the velocity of the ball at $t = 3$ seconds?
If a ball's height is given by $h(t) = -12t^2 + 75t$, what is the velocity of the ball at $t = 3$ seconds?
Determine the derivative of the function $f(x) = 2x(x^2 + 4)$.
Determine the derivative of the function $f(x) = 2x(x^2 + 4)$.
Find the derivative of $y = x^3 - 5x^{-2}$.
Find the derivative of $y = x^3 - 5x^{-2}$.
Simplify the expression $\frac{x^n}{x^m}$
Simplify the expression $\frac{x^n}{x^m}$
Simplify the expression $(x^n)^m$
Simplify the expression $(x^n)^m$
Flashcards
Average Rate of Change
Average Rate of Change
The slope of the secant line between two points on a curve. Calculated as (f(b) - f(a)) / (b - a).
Instantaneous Rate of Change
Instantaneous Rate of Change
The slope of the tangent line at a single point on a curve. It represents the instantaneous rate of change of the function.
Constant Function Rule
Constant Function Rule
The derivative of a constant function is always zero.
Power Rule
Power Rule
Signup and view all the flashcards
Constant Multiple Rule
Constant Multiple Rule
Signup and view all the flashcards
Sum and Difference Rule
Sum and Difference Rule
Signup and view all the flashcards
Product Rule
Product Rule
Signup and view all the flashcards
Quotient Rule
Quotient Rule
Signup and view all the flashcards
Power of a Function Rule
Power of a Function Rule
Signup and view all the flashcards
Chain Rule
Chain Rule
Signup and view all the flashcards
Study Notes
Differentiation Overview
- Differentiation is the rate of change, slope of a tangent line, and utilizes rules of differentiation
Average Rate of Change
- The average rate of change is the slope of the secant line
- Formula: Δy/Δx = (f(b) – f(a))/(b-a)
Instantaneous Rate of Change
- The instantaneous rate of change is the slope of the tangent line
- Formula: dy/dx = lim h→0 (f(a + h) – f(a))/h
- The slope of the tangent line is the same as the derivative of the function.
Rules of Differentiation
- Constant Function: If f(x) = c, then f'(x) = 0
- Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
- Constant Multiple Rule: If y = c * f(x), then y' = c * f'(x)
- Sum and Difference Rule: If y = f + g, then y' = f' + g'
- Product Rule: If y = f * g, then y' = f * g' + g * f'
- Quotient Rule: If y = f/g, then y' = (gf' - fg') / g^2
- Power of a Function Rule: If y = [g(x)]^n, then y' = n[g(x)]^(n-1) * g'(x)
- Chain Rule: If F(x) = f(g(x)), then F'(x) = f'(g(x)) * g'(x)
- Leibniz Form of Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx)
Laws of Exponents (Recall)
- x^n * x^m = x^(n+m)
- x^n / x^m = x^(n-m)
- (x^n)^m = x^(nm)
- (ab)^n = a^n * b^n
- (a/b)^n = a^n / b^n
- x^0 = 1
- x^(-n) = 1 / x^n
- x^(m/n) = nth root of (x^m)
Guided Practice Examples
- Finding the average rate of change for f(x) = 4x^3 – 5x from x = 0 to x = 2 yields a result of 11.
- Finding the average rate of change for y = x(x – 1)^2 from x = -1 to x = 2 yields a result of 2.
- The slope of the line tangent to y = x^3 – 5x^(-2) is y' = 3x^2 + 10/x^3
- The slope of the line tangent to f(x) = 2x(x^2 + 4) is f'(x) = 2(3x^2 + 4)
- The slope of the tangent line for f(x) = 4x^2 – 7 at point (3,5) is m = 24.
- The slope of the tangent line for y = −2x^2(x − 3) at x = 1 is m = 6.
- Given a ball's height with respect to time: h(t) = -12t^2 + 75t (h in feet, t in seconds.) The ball is moving at 3 ft/sec at t = 3 seconds.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
