Rates, Tables, and Particle Motion

Mean Value Theorem

or

Intermediate Value Theorem

Average Value

Average Rate of Change

Integral

Since $$f$$ is Continous and differentiable, the Mean Value Theorem can be applied

Since $$f$$ is Continous, the Mean Value Theorem can be applied

Since $$f$$ is increasing, the Mean Value Theorem can be applied

Since $$f$$ is differentiable, the Mean Value Theorem can be applied

Since $$f$$ is Continous and differentiable, the Intermediate Value Theorem can be applied

Since $$f$$ is Continous, the Intermediate Value Theorem can be applied

Since $$f$$ is increasing, the Intermediate Value Theorem can be applied

Since $$f$$ is differentiable, the Intermediate Value Theorem can be applied

Identify  $$f\left(x\right)$$  , then find  $$f'\left(x\right)$$  

Factor and cancel

Compare limits to infinity

Identify  $$f\left(x\right)$$  

$$y-y_1=m\left(x-x_1\right)$$

Find $$f'\left(a\right)$$

$$y=mx+b$$

$$ax+by=c$$

$$\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

$$f'\left(a\right)$$

$$y-y_1=m\left(x-x_1\right)$$

IVT. Write "Since  $$f\left(x\right)$$  is continuous, the IVT applies.  Since...(lower), and ....(higher), there exists a c such that  $$f\left(c\right)=L$$  

Tangent line.  $$y-y_1=m\left(x-x_1\right)$$  

IVT. Write since...(lower) and ...(higher), there exists a c such that  $$f\left(c\right)=L$$