**MATH 115 REVIEW PROBLEMS**

The following are review exercises for the Math 115 final exam. These exercises are provided for you to practice or test yourself for readiness for the final exam. Your final exam will be in two parts: the first part does not allow the use of a calculator, and the second part does allow the use of a graphing calculator. Please note that for the final, you may use any graphing calculator except the TI-89, TI-Nspire, and any calculator with a QWERTY keypad. Show all your work: unsupported results may not receive credit.

**1.** Graph *f*(*x*) = *x*^{2}+*x* and *g*(*x*) = 2*x*+2 on the same set of coordinate axes. Use your graphs to solve the equation *f*(*x*) = *g*(*x*). Solve the inequality *f*(*x*) > *g*(*x*) and express your answer using interval notation.

**2.** Suppose *g* is defined on the restricted domain (-4,-2] by the formula *g*(*x*) = (*x*+1)^{4}. Graph *y* = *g*(*x*). Solve the equation *g*(*x*) = 25. Find *g*^{-1}(16). Find a formula for *g*^{-1}(*x*).

**3.** Sketch a graph of *f*(*x*) = (*x* -5)/*(x*^{ 2} -2*x* -3). Identify the *x*- and *y*- intercepts. Give the equations of the horizontal and vertical asymptotes. Find the domain of log[(*x* -5)/*(x*^{ 2} -2*x* -3)].

**4.** Graph *f*(*x*) = *e*^{x+1}-4. Identify the asymptote(s). Find the *x*-intercepts and *y*-intercepts exact. Estimate the intercepts accurate to 2 decimal places. Find *f*^{ -1}(*x*).

**5.** Graph *f*(*x*) = ln(*x*+3)-1. Identify the asymptote(s). Find the *x*-intercepts and *y*-intercepts exact. Estimate the intercepts accurate to 2 decimal places. Find *f*^{ -1}(*x*).

**6.** Let *g*(*x*) = 2ln(*x*)+3ln(*x*+4). Find the domain of *g*. Graph y=*g*(*x*). Identify the asymptote(s). Identify the intercept(s) accurate to 2 decimal places. Solve *g*(*x*) = 1 accurate to one decimal place.

**7.(a)** Solve 2^{3-x} = 5^{2x+1}. Give your answers both exact and a numerical approximation accurate to 2 decimal places.

** (b).** Solve for *x* EXACT: 3*e ^{x}*-

*x*

^{2}

*e*

^{x}=0.** (c).** Solve for *x* accurate to two decimal places for *x* in the interval (0,p): sin(4*x*) = ln(*x*+2).

**8.** The point *A* is on the unit circle *U* at the right. The arc length from *E* counterclockwise around *U* to *A* is 4. Mark *A* accurately on *U*. In what quadrant is *A*? What are the coordinates (*x*,*y*) of *A* accurate to 2 decimal places?

**9.** Graph one period of *y* = -2cos(2*x*-p/6). Label all *x*-intercepts, maximum, and minimum points exactly. What is the period, amplitude, and phase shift?

**10.** Suppose you put 5,000 dollars into an investment that earns 7% interest compounded continuously, and at the same time your friend invests 4,000 dollars in an investment that earns 9% interest compounded monthly. Which account will be worth MORE at 14 years? JUSTIFY YOUR ANSWER

**11.** The population of a certain town grows from 10,000 in 1982 to 35,000 in 1995. At this rate, in what year will the population reach 300,000?

**12.** The half-life of a radioactive isotope is 1250 years. Starting with 100 grams of the isotope, how much will remain after 150 years?

**13.** If q = *arcsin*(*x*), express *tan*(q) as a function of *x*.

**14.** A forest ranger is located in an observation tower and notices two fires at distances of 4 and 7 miles, respectively, from the tower. If the angle between the lines of sight to the two fire points is 137^{o}, how far apart are the fires?

**15.** Sitting in Rutgers stadium, Josh notices a blimp straight ahead and above him at an angle of elevation of 58^{o}. Three minutes later, he notices the blimp is still straight ahead, but now at an angle of elevation of 48^{o}. If the blimp maintained an altitude of 2000 feet, how far did the blimp travel in those three minutes?

** DEFINITELY Review the material covered in the
Handout: The Language of Functions
**

** ALSO definitely check out the final exam reviews on the Math 111 and Math 112 home pages. **