Alternate proof of product property Assume that y > 0 is

Integrals with general bases Evaluate the following integrals. \(\int_{-1}^{1} 10^{x} d x\)

Integrals with \(e^{x}\) Evaluate the following integrals \(\int_{-2}^{2} \frac{e^{x / 2}}{e^{x / 2}+1} d x\)

Integrals with \(e^{x}\) Evaluate the following integrals \(\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x\)

Derivatives with ln x Evaluate the following derivatives \(\frac{d}{d x}(\sin (\ln x))\)

Derivatives with ln x Evaluate the following derivatives \(\frac{d}{d x}(\ln (\ln x))\)

Derivatives with ln x Evaluate the following derivatives \(\left.\frac{d}{d x}\left(x \ln \left(x^{3}\right)\right)\right|_{x=1}\)

What is the domain and range of ln x?

Give a geometrical interpretation of the function

Evaluate \(\int 4^{x} d x\)

What is the inverse function of and what is its domain and range?

Express , , using the base e.

Evaluate

Derivatives with ln x Evaluate the following derivatives.

Integrals with ln x Evaluate the following integrals. Include absolute values only when needed.

Integrals with ln x Evaluate the following integrals. Include absolute values only when needed.

Integrals with ln x Evaluate the following integrals. Include absolute values only when needed.

Integrals with ln x Evaluate the following integrals. Include absolute values only when needed.

Integrals with ex Evaluate the following integrals.

Integrals with ex Evaluate the following integrals.

Integrals with general bases Evaluate the following integrals.

Integrals with general bases Evaluate the following integrals.

Integrals with general bases Evaluate the following integrals.

Derivatives Evaluate the derivatives of the following functions.

Derivatives Evaluate the derivatives of the following functions.

Derivatives Evaluate the derivatives of the following functions.

Derivatives Evaluate the derivatives of the following functions.

Derivatives Evaluate the derivatives of the following functions.

Derivatives Evaluate the derivatives of the following functions.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0. a. In (xy) = In x + In y b. In 0 = 1 c. In (x + y) = In x + In y d. \(2^{x}=e^{2 \ln x}\) e. The area under the curve y = 1/x and the x-axis on the interval [1, e] is 1.

Logarithm properties Use the integral definition of the natural logarithm to prove that ln (x/y) = ln x- ln y

Calculator limits Use a calculator to make a table similar to Table 6.2 to approximate the following limits. Confirm your result with l'Hopital's Rule \(\lim _{h \rightarrow 0}(1+2 h)^{1 / h}\)

Calculator limits Use a calculator to make a table similar to Table 6.2 to approximate the following limits. Confirm your result with l'Hopital's Rule \(\lim _{h \rightarrow 0}(1+3 h)^{2 / h}\)

Calculator limits Use a calculator to make a table similar to Table 6.2 to approximate the following limits. Confirm your result with l'Hopital's Rule \(\lim _{x \rightarrow 0} \frac{2^{x}-1}{x}\)

Calculator limits Use a calculator to make a table similar to Table 6.2 to approximate the following limits. Confirm your result with l'Hopital's Rule \(\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}\)

Zero net area Consider the function \(f(x)=\frac{1-x}{x}\) a. Are there numbers 0 <a <1 such that \(\int_{1-a}^{1+a} f(x) d x=0 ?\) b. Are there numbers a > 1 such that \(\int_{1 / a}^{a} f(x) d x=0 ?\)

Behavior at the origin Using calculus and accurate sketches, explain how the graphs of \(f(x)=x^{p}\) differ as \(x \rightarrow 0\) for \(p=\frac{1}{2}, 1,2\)

Average value What is the average value of f(x)=1/x on the interval [1,p] for p>1? What is the average value of f as \(p \rightarrow \infty\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(x^{2 x}\right)\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(e^{-10 x^{2}}\right)\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(x^{\tan x}\right)\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left[\left(\frac{1}{x}\right)^{x}\right]\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(x^{e}+e^{x}\right)\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(1+\frac{4}{x}\right)^{x}\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right)\)

Miscellaneous derivatives Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(\cos \left(x^{2 \sin x}\right)\right)\)

Miscellaneous integrals Evaluate the following integrals. \(\int 7^{2 x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int 3^{-2 x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{5} 5^{5 x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int x^{2} 10^{x^{3}} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{\pi} \cos x \cdot 2^{\sin x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int_{1}^{2 e} \frac{3^{\ln x}}{x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int \frac{\sin (\ln x)}{4 x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int_{1}^{e^{2}} \frac{(\ln x)^{5}}{x} d x\)

Miscellaneous integrals Evaluate the following integrals. \(\int \frac{\ln ^{2} x+2 \ln x-1}{x} d x\)

Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one fourth the area of the square? Begin by showing that x and y must satisfy xy < \(\frac{1}{2}\) in order for the area condition to be met. Then argue that the required probability \(\frac{1}{2}+\int_{1 / 2}^{1} \frac{d x}{2 x}\) and evaluate the integral.

Derivative of ln |x| Differentiate ln x for x>0 and differentiate ln(-x) for x<0 to conclude that \(\frac{d}{d x}(\ln |x|)=\frac{1}{x}\)

Properties of \(e^{x}\) Use the inverse relations between ln x and \(e^{x}\) and the properties of ln x to prove the following properties. a. \(e^{x-y}=\frac{e^{x}}{e^{y}}\) b. \(\left(e^{x}\right)^{y}=e^{x y}\)

In x is unbounded Use the following argument to show that \(\lim _{x \rightarrow \infty} \ln x=\infty\)and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\). a. Make a sketch of the function f(x) = 1/x on the interval [1,2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1,2] is in 2. b. Construct a rectangle over the interval [1, 2] with height \(\frac{1}{2}\). Explain why In 2 > \(\frac{1}{2}\) c. Show that in \(2^{n}\) > n/2 and In \(2^{-n}\) <-n/2. d. Conclude that lim \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\).

Bounds on e Use a left Riemann sum with at least n = 2 subintervals of equal length to approximate \(\ln 2=\int_{1}^{2} \frac{d t}{t}\) show that in 2 < 1. Use a right Riemann sum with at least n = 7 subintervals of equal length to approximate \(\ln 3=\int_{1}^{3} \frac{d t}{t}\) and show that In 3 > 1.

Harmonic sum In Chapter 8, we will encounter the harmonic sum \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\) Use a right Riemann sum to approximate \(\int_{1}^{n} \frac{d x}{x}\) (with unit spacing between the grid points) to show that \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}>\ln (n+1)\). Use this fact to conclude that \(\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)\) does not exist.

Alternate proof of product property Assume that y>0 is fixed and that x > 0. Show that \(\frac{d}{d x}(\ln x y)=\frac{d}{d x}(\ln x)\). Recall that if two functions have the same derivative, they differ by a constant. Set x = 1 to evaluate the constant and prove that In xy = In x + In y.