91影视

Show that $y\left(x\right)=3\left(2^x\right)$is a solution of the differential equation $\frac{dy}{dx}=\left(\ln \left(2\right)\right)y$

Finding antiderivatives by undoing the chain rule: For each function f that follows, find a function F with the property that F(x) = f(x). You may have to guess and check to find such a function

Translate expressions written in Leibniz notation to 鈥減rime鈥� notation, and vice versa.

$\frac{d}{dx}\left(x^2\right)$

Find the derivatives of the function:$f\left(x\right)=\frac{(x-2)}{(x-1)(x+3)}.$

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible:

what it means for a function $f$ to be differentiable on an open or closed interval I

Explain why the power rule cannot be used to differentiate the function${(2-x)}^{\frac{1}{3}}$

The derivatives of the function $f\left(x\right)=\cos \left(3x^2\right)$) that follow are incorrect. What misconception occurs in each case?

$$\begin{gathered} \text{(a) Incorrect:}{f}^{\text{'}}\left(x\right)=-\sin \left(3x^2\right)\text{.} \\ \text{(b) Incorrect:}{f}^{\text{'}}\left(x\right)=-\sin \left(3x^2\right)\left(6x\right)\left(6\right)\text{.} \end{gathered}$$

The function $f\left(x\right)=4-x^2$is both continuous and differentiable at x = 1. Write these facts as limit statements.

On a graph of f(x) = $x^2$,

(a) draw the tangent line to the graph of f at the point (2, f(2));

(b) draw the secant line from (2, f(2)) to (2.75, f(2.75));

(c) draw the secant line from (1.75, f(1.75)) to (2, f(2)).

(d) Which secant line is a better approximation to the tangent line, and why?