3 Rates of Change and Derivatives Free Online FlashCards

What is the average rate of change from $x=a$ to $x=b$?

The average rate of change is $\frac{f(b)-f(a)}{b-a}$, representing output change divided by input change over an interval.

What does an instantaneous rate of change represent?

It is the slope of the tangent line to the graph at that point, describing how the quantity changes at one particular input value.

State the limit definition of $f'(a)$.

The derivative at $a$ is $f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$, when this limit exists.

Why is $h$ not set equal to zero in the derivative quotient?

The quotient cannot be evaluated at $h=0$ because it would require division by zero. The limit instead examines values as nonzero $h$ gets arbitrarily close to zero.

Using the definition, what is the derivative of $f(x)=x^2$?

For $f(x)=x^2$, the definition gives $f'(x)=2x$. Therefore, $f'(3)=6$.

How are the units of a derivative determined?

The derivative has units of output divided by input: $\text{units of }f'=\frac{\text{units of }f}{\text{units of }x}$.

How are velocity and acceleration related to position?

If $s(t)$ is position, then instantaneous velocity is $v(t)=s'(t)$, and instantaneous acceleration is $a(t)=s''(t)$.

Find $v(2)$ and $a(2)$ for $s(t)=t^2+3t$.

For $s(t)=t^2+3t$, $v(t)=2t+3$ and $a(t)=2$. Thus, at $t=2$, velocity is $7\text{ m/s}$ and acceleration is $2\text{ m/s}^2$.

What is the power rule?

The power rule is $\frac{d}{dx}(x^n)=nx^{n-1}$ for a real number $n$.

What is the derivative of a constant?

The derivative of a constant is zero: if $f(x)=C$, then $f'(x)=0$.

How do sum and difference rules simplify differentiation?

Differentiate each term separately. For $f(x)=3x^4-5x^2+7$, the derivative is $f'(x)=12x^3-10x$.

How do velocity and acceleration indicate speeding up or slowing down?

An object speeds up when velocity and acceleration have the same sign; it slows down when they have opposite signs.