Differentiate $y=e^{x+1}+1$.

Consider $f(x)=x^{5 / 2} e^x$. Find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$.

Consider $y=\dfrac{\left(x^2+1\right)^4}{(2 x+1)^3(3 x-1)^5}$. Calculate $y^{\prime}$.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Estimate the rate of growth in 1985.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Use a model for the rate of population growth in the 20th century to estimate the rates of growth in 1920 and 1980. Compare with the estimates by averaging the slopes of two secant lines.

Suppose that the cost (in dollars) for a company to produce $x$ pairs of a new line of jeans is

$$C(x)=2000+3 x+0.01 x^2+0.0002 x^3$$

Compare $C^{\prime}(100)$ with the cost of manufacturing the 101st pair of jeans.

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

$$x^2+y^2=\left(2 x^2+2 y^2-x\right)^2, \quad\left(0, \frac{1}{2}\right) \quad (\text{cardioid})$$

Suppose that the cost (in dollars) for a company to produce $x$ pairs of a new line of jeans is

$$C(x)=2000+3 x+0.01 x^2+0.0002 x^3$$

Find $C^{\prime}(100)$ and, using correct units, explain the meaning of your answer in the context of this problem.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Use a cubic function model to find a model for the rate of population growth in the 20th century.

Consider $x e^y=y-1$. Calculate $y^{\prime}$.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Use a graphing calculator or computer to find a cubic function (a third-degree polynomial) that models the data.

Suppose that the cost (in dollars) for a company to produce $x$ pairs of a new line of jeans is

$$C(x)=2000+3 x+0.01 x^2+0.0002 x^3$$

Find the marginal cost function.

The table gives the population of the world in the 20th century.

$$\begin{array}{|c|c||c|c|} \hline \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} & \text { Year } & \begin{array}{c} \text { Population } \\ \text { (in millions) } \end{array} \\ \hline 1900 & 1650 & 1960 & 3040 \\ 1910 & 1750 & 1970 & 3710 \\ 1920 & 1860 & 1980 & 4450 \\ 1930 & 2070 & 1990 & 5280 \\ 1940 & 2300 & 2000 & 6080 \\ 1950 & 2560 & & \\ \hline \end{array}$$

Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.

Let $y=\sqrt{x}$. Sketch a diagram showing the line segments with lengths $d x, d y$, and $\Delta y$.