미분법 전체 문제

[T] Using the tables of first and second derivatives and the best fit, answer the following questions:

a. Will the model be accurate in predicting the future population of New York City? Why or why not? b. Estimate the population in 2010. Was the prediction correct from a.?

T/F: The Product Rule states that $\displaystyle \frac{d}{dx}\big(x^2\sin x\big) = 2x\cos x$.

T/F: The Quotient Rule states that $\displaystyle \frac{d}{dx}\left(\frac{x^2}{\sin x}\right) = \frac{\cos x}{2x}$.

T/F: The derivatives of the trigonometric functions that start with “c” have minus signs in them.

What derivative rule is used to extend the Power Rule to include negative integer exponents?

T/F: Regardless of the function, there is always exactly one right way of computing its derivative.

In your own words, explain what it means to make your answers “clear.”

T/F: The Chain Rule describes how to evaluate the derivative of a composition of functions.

T/F: The Generalized Power Rule states that $\displaystyle \frac{d}{dx}\Big( g(x)^n\Big) = n\big(g(x)\big)^{n-1}$.

T/F: $\displaystyle \frac{d}{dx}\big(\ln (x^2)\big) = \frac{1}{x^2}$.

T/F: $\displaystyle \frac{d}{dx}\big(3^x\big) \approx 1.1\cdot3^x$.

T/F: $\displaystyle \frac{dx}{dy} = \frac{dx}{dt}\cdot \frac{dt}{dy}$

$f(x) = \big(\ln x + x^2\big)^3$

In your own words, explain the difference between implicit functions and explicit functions.

Implicit differentiation is based on what other differentiation rule?

T/F: Implicit differentiation can be used to find the derivative of $y=\sqrt{x}$.

T/F: Implicit differentiation can be used to find the derivative of $y=x^{3/4}$.

T/F: Every function has an inverse.

In your own words explain what it means for a function to be “one to one.”

If $(1,10)$ lies on the graph of $y=f(x)$, what can be said about the graph of $y=f^{-1}(x)$?