Find df/dx if

$$f(x)=\int_{1}^{e^{x}} \frac{2 \ln t}{t} d t.$$

Use finite approximations to estimate the area under the graph of the function using a lower sum with four rectangles of equal width. $f(x)=1 / x$ between x = 1 and x = 5.

Calculators have taken some of the mystery out of this once-challenging question. (Go ahead and check; you will see that it is a surprisingly close call.) You can answer the question without a calculator, though.

a. Find an equation for the line through the origin tangent to the graph of $y=\ln x$.

b. Give an argument based on the graphs of $y=\ln x$ and the tangent line to explain why $\ln x<x / e$ for all positive $x \neq e$.

c. Show that $\ln \left(x^e\right)<x$ for all positive $x \neq e$.

d. Conclude that $x^e<e^x$ for all positive $x \neq e$.

e. So which is bigger, $\pi^e$ or $e^\pi$ ?

Suppose that $\sum_{k=1}^{10} a_{k}=-2$ and $\sum_{k=1}^{10} b_{k}=25.$ Find the value of

$$\sum_{k=1}^{10}\left(b_{k}-3 a_{k}\right)$$

Find a formula for the Riemann sum obtained by dividing the interval [a, b] into n equal subintervals and using the right-hand endpoint for each $c_{k}.$ Then take a limit of these sums as $n \rightarrow \infty$ to calculate the area under the curve over [a, b]. $f(x)=1-x^{2}$ over the interval [0, 1].

The linearization of $\log _3 x$

a. Find the linearization of $f(x)=\log _3 x$ at $x=3$. Then round its coefficients to two decimal places.

b. Graph the linearization and function together in the windows $0 \leq x \leq 8$ and $2 \leq x \leq 4$.

Evaluate the integrals. $\int_{-\pi / 4}^{0} \tan x \sec ^{2} x\ d x$

Graph each function f(x) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum $\sum\limits_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},$ given that $c_{k}$ is the midpoint of the kth subinterval. (Make a separate sketch for each set of rectangles.) Find the norm of the partition $P = \{-2, -1.6, -0.5, 0, 0.8, 1\}$.

The linearization of $2^x$

a. Find the linearization of $f(x)=2^x$ at $x=0$. Then round its coefficients to two decimal places.

b. Graph the linearization and function together for $-3 \leq x \leq 3$ and $-1 \leq x \leq 1$.

Could $x^{\ln 2}$ possibly be the same as $2^{\ln x}$ for some x > 0? Graph the two functions and explain what you see.

Graph each function f(x) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum $\sum\limits_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},$ given that $c_{k}$ is the right-hand endpoint (Make a separate sketch for each set of rectangles.) Find the norm of the partition $P = \{-2, -1.6, -0.5, 0, 0.8, 1\}$.

In this exercise, find the derivative of y with respect to x, t, or $\theta$, as appropriate.

$$y=\ln \left(3 \theta e^{-\theta}\right)$$

Graph each function f(x) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum $\sum\limits_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},$ given that $c_{k}$ is the left-hand endpoint. (Make a separate sketch for each set of rectangles.) Find the norm of the partition $P = \{-2, -1.6, -0.5, 0, 0.8, 1\}$.

What do the conclusions we drew in the lessons from the second section, about the limits of rational functions tell us about the relative growth of polynomials as $x \rightarrow \infty$ ?