 Calculus Volume 1

# Review Exercises

Calculus Volume 1Review Exercises

### Review Exercises

True or False? Justify your answer with a proof or a counterexample.

310 .

A function is always one-to-one.

311 .

$f∘g=g∘f,f∘g=g∘f,$ assuming f and g are functions.

312 .

A relation that passes the horizontal and vertical line tests is a one-to-one function.

313 .

A relation passing the horizontal line test is a function.

For the following problems, state the domain and range of the given functions:

$f=x2+2x−3,g=ln(x−5),h=1x+4f=x2+2x−3,g=ln(x−5),h=1x+4$

314 .

h

315 .

g

316 .

$h∘fh∘f$

317 .

$g∘fg∘f$

Find the degree, y-intercept, and zeros for the following polynomial functions.

318 .

$f(x)=2x2+9x−5f(x)=2x2+9x−5$

319 .

$f(x)=x3+2x2−2xf(x)=x3+2x2−2x$

Simplify the following trigonometric expressions.

320 .

$tan 2 x sec 2 x + cos 2 x tan 2 x sec 2 x + cos 2 x$

321 .

$cos 2 x - sin 2 x cos 2 x - sin 2 x$

Solve the following trigonometric equations on the interval $θ=[−2π,2π]θ=[−2π,2π]$ exactly.

322 .

$6cos2x−3=06cos2x−3=0$

323 .

$sec 2 x − 2 sec x + 1 = 0 sec 2 x − 2 sec x + 1 = 0$

Solve the following logarithmic equations.

324 .

$5 x = 16 5 x = 16$

325 .

$log 2 ( x + 4 ) = 3 log 2 ( x + 4 ) = 3$

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse $f−1(x)f−1(x)$ of the function. Justify your answer.

326 .

$f(x)=x2+2x+1f(x)=x2+2x+1$

327 .

$f ( x ) = 1 x f ( x ) = 1 x$

For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.

328 .

$f ( x ) = 9 − x f ( x ) = 9 − x$

329 .

$f ( x ) = x 2 + 3 x + 4 f ( x ) = x 2 + 3 x + 4$

330 .

A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and$1000 for 100 shirts.

331 .

a. Find the equation $C=f(x)C=f(x)$ that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each. 332 . a. Find the inverse function $x=f−1(C)x=f−1(C)$ and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has$8000 to spend.

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

333 .

The population can be modeled by $P(t)=82.5−67.5cos[(π/6)t],P(t)=82.5−67.5cos[(π/6)t],$ where $tt$ is time in months $(t=0(t=0$ represents January 1) and $PP$ is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

334 .

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as $P(t)=82.5−67.5cos[(π/6)t]+t,P(t)=82.5−67.5cos[(π/6)t]+t,$ where $tt$ is time in months ($t=0t=0$ represents January 1) and $PP$ is population (in thousands). When is the first time the population reaches 200,000?

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation $y=ert,y=ert,$ where $yy$ is the ratio of radiocarbon still present in the material, $tt$ is the number of years passed, and $r=−0.0001210r=−0.0001210$ is the decay rate of radiocarbon.

335 .

If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

336 .

Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?

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