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Study Guides > Calculus Volume 1

Chapter 1 Review Exercises

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

1. A function is always one-to-one.

2. f∘g=g∘ff \circ g=g\circ f, assuming ff and gg are functions.

Answer:

False

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

4. A relation passing the horizontal line test is a function.

Answer:

False

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

f=x2+2x−3,g=ln⁡(x−5),h=1x+4f=x^2+2x-3,\phantom{\rule{3em}{0ex}}g=\ln(x-5),\phantom{\rule{3em}{0ex}}h=\frac{1}{x+4}

5. hh

6. gg

Answer:

Domain: x>5x>5, Range: all real numbers

7. h∘fh\circ f

8. g∘fg\circ f

Answer:

Domain: x>2x>2 or x<−4x<-4, Range: all real numbers

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

9. f(x)=2x2+9x−5f(x)=2x^2+9x-5

10. f(x)=x3+2x2−2xf(x)=x^3+2x^2-2x

Answer:

Degree of 3, yy-intercept: 0, Zeros: 0, 3−1, −1−3\sqrt{3}-1, \, -1-\sqrt{3}

Simplify the following trigonometric expressions.

11. tan⁡2xsec⁡2x+cos⁡2x\frac{\tan^2 x}{\sec^2 x}+\cos^2 x

12. cos⁡(2x)=sin⁡2x \cos(2x)=\sin^2 x

Answer:

cos⁡(2x) \cos(2x) or 12(cos⁡(2x)+1)\frac{1}{2}(\cos(2x)+1)

Solve the following trigonometric equations on the interval θ=[−2π,2π]\theta =[-2\pi ,2\pi] exactly.

13. 6cos⁡2x−3=06\cos^2 x-3=0

14. sec⁡2x−2sec⁡x+1=0\sec^2 x-2\sec x+1=0

Answer:

0, ±2π0, \, \pm 2\pi

Solve the following logarithmic equations.

15. 5x=165^x=16

16. log⁡2(x+4)=3\log_2 (x+4)=3

Answer:

4

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.

17. f(x)=x2+2x+1f(x)=x^2+2x+1

18. f(x)=1xf(x)=\frac{1}{x}

Answer:

One-to-one; yes, the function has an inverse; inverse: f−1(x)=1xf^{-1}(x)=\frac{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.

19. f(x)=9−xf(x)=\sqrt{9-x}

20. f(x)=x2+3x+4f(x)=x^2+3x+4

Answer:

x≥−32, f−1(x)=−32+124y−7x \ge -\frac{3}{2}, \, f^{-1}(x)=-\frac{3}{2}+\frac{1}{2}\sqrt{4y-7}

21. 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∘55^{\circ} 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.

22. 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.

Answer:

a. C(x)=300+7xC(x)=300+7x b. 100 shirts

23. 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.

24. The population can be modeled by P(t)=82.5−67.5cos⁡[(π/6)t]P(t)=82.5-67.5\cos [(\pi /6)t], where tt is time in months (t=0t=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?

Answer:

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

25. 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]+tP(t)=82.5-67.5\cos [(\pi /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=erty=e^{rt}, where yy is the percentage 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.

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

Answer:

78.51%

27. 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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