Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.

$$\left|\begin{array}{cc}{e^{2 x}} & {e^{3 x}} \\ {2 e^{2 x}} & {3 e^{3 x}}\end{array}\right|$$

Find the determinant of the matrix. Expand by cofactors on each indicated row or column.

$$\left[\begin{array}{cccc}{10} & {8} & {3} & {-7} \\ {4} & {0} & {5} & {-6} \\ {0} & {3} & {2} & {7} \\ {1} & {0} & {-3} & {2}\end{array}\right]$$

(a) Row 3, (b) Column 1

$$\text { let } \mathbf { v } = \langle 1,3 , - 2 \rangle \text { and } \mathbf { w } = \langle 2 , - 1,4 \rangle$$

$$\text { Compute } \mathbf { v } \times \mathbf { w }$$

Regroup. Write the missing numbers. ___ tens 15 ones = 6 tens 5 ones

Which is the first incorrect step in simplifying $\log_9 243?$

$$\begin{array}{ll}{\text { Step } 1: \log _{9} 243} {=x} \\ {\text { Step } 2:}\ {9^{x}=243} \\ {\text { Step } 3:}\ {x=243 \div 9} \\ {\text { Step } 4:}\ {=27}\end{array}$$

A. Step 1 B. Step 2 C. Step 3 D. Step 4

The daily oxygen consumption of a bird embryo increases from the time the egg is laid through the time the chick hatches. In a typical galliform bird, the oxygen consumption can be approximated by $c(t)=-0.065 t^{3}+3.4 t^{2}-22 t+3.6$ ml $(8 \leq t \leq 30)$ where t is the time (in days) since the egg was laid. (An egg will typically hatch at around t=28.) Use the model to estimate the following (give the units of measurement for each answer and round all answers to two significant digits): a. The daily oxygen consumption 20 days after the eggwas laid b. The rate at which the oxygen consumption is changing 20 days after the egg was laid c. The rate at which the oxygen consumption is accelerating 20 days after the egg was laid

Evaluate the following limits using Taylor series. $\lim _ { x \rightarrow 0 } \frac { e ^ { x } - 1 } { x }$

An airplane wing is designed so that the speed of the air across the top of the wing is 251 m/s when the speed of the air below the wing is 225 m/s. The density of the air is

$$1.29 \mathrm { kg } / \mathrm { m } ^ { 3 }$$

What is the lifting force on a wing of area

$$24.0 \mathrm { m } ^ { 2 } ?$$

Use matrices A, B, C, and D to find each scalar product and sum, or difference, if possible. If an operation is not defined, label it undefined.

$$A=\left[\begin{array}{rrrr}{6} & {1} & {0} & {8} \\ {-4} & {3} & {7} & {11}\end{array}\right] \quad B=\left[\begin{array}{rr}{1} & {3} \\ {-2} & {4}\end{array}\right] \quad C=\left[\begin{array}{rr}{-2} & {1} \\ {4} & {0} \\ {2} & {2} \\ {1} & {1}\end{array}\right] \quad D=\left[\begin{array}{rr}{5} & {-2} \\ {3} & {6}\end{array}\right]$$

B - 2A

(a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one? (c) Suppose that f is a one-to-one function with domain A and range B. How is the inverse function

$$f ^ { - 1 }$$

defined? What are the domain and range of

$$f ^ { - 1 }$$

? (d) If you are given a formula for f, how do you find a formula for

$$f ^ { - 1 }$$

? Find the inverse of the function f(x) = 2x. (e) If you are given a graph of f, how do you find a graph of the inverse function

$$f ^ { - 1 }$$

?

Solve by factoring. z² - 10z + 24 = 0

Express the first trigonometric function in terms of the second. $\tan \theta, \cos \theta$

Use Heron’s Area Formula to find the area of the triangle. $a=64.8, \quad b=49.2, \quad c=24.1$

A motorist driving a 1200 kg car on level ground accelerates from 20.0 m/s to 30.0 m/s in a time of 5.0 s. Ignoring friction and air resistance, determine the average mechanical power in watts the engine must supply during this time interval.