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Precalculus Review Problems

Introduction

Precalculus serves as the bridge between algebra and calculus, providing essential mathematical tools and concepts. This review covers fundamental areas including functions, temperature conversion, motion problems, and cost functions. Mastering these topics will build a strong foundation for calculus and higher mathematics.

Functions

A function is a relationship where each input has exactly one output. We express functions using notation such as f(x), where f is the name of the function and x is the input.

Example 1: Given f(x) = 3x - 2x + 1, find f(2).

Solution: Substitute x = 2 into the function: f(2) = 3(2) - 2(2) + 1 = 3(4) - 4 + 1 = 12 - 4 + 1 = 9

Example 2: If f(x) = 2x + 3 and g(x) = x - 1, find (f g)(x), the composition of f and g.

Solution: (f g)(x) = f(g(x)) = f(x-1) = 2(x-1) + 3 = 2x - 2 + 3 = 2x + 1

Types of Functions

  • Polynomial Functions: Functions with terms involving non-negative integer powers of x, such as f(x) = x + 3x + 2
  • Rational Functions: Ratios of polynomial functions, such as f(x) = (x+1)/(x-2)
  • Exponential Functions: Functions where the variable is in the exponent, such as f(x) = e^x or f(x) = 2^x
  • Logarithmic Functions: Inverse functions of exponential functions, such as f(x) = log(x) or f(x) = ln(x)

Temperature Conversion

Temperature conversion problems involve changing temperature values between different scales, primarily Celsius (C) and Fahrenheit (F), and occasionally Kelvin (K).

Celsius to Fahrenheit: F = (9/5)C + 32

Fahrenheit to Celsius: C = (5/9)(F - 32)

Celsius to Kelvin: K = C + 273.15

Example 1: Convert 25C to Fahrenheit.

Solution: F = (9/5)(25) + 32 = 45 + 32 = 77F

Example 2: The boiling point of water is 212F. What is this temperature in Celsius?

Solution: C = (5/9)(212 - 32) = (5/9)(180) = 100C

Example 3: Find the temperature where Celsius and Fahrenheit readings are the same.

Solution: Set C = F in the conversion formula: F = (9/5)F + 32
F - (9/5)F = 32
(-4/5)F = 32
F = -40
Therefore, -40C = -40F

Motion Problems

Motion problems involve calculating distance, speed, velocity, or acceleration using fundamental equations of motion.

Distance: d = r t (where r is rate of speed and t is time)

Displacement: s = s + vt + (1/2)at (where s is initial position, v is initial velocity, and a is acceleration)

Velocity: v = v + at

Example 1: A car travels at an average speed of 65 mph for 3.5 hours. How far does it travel?

Solution: d = r t = 65 3.5 = 227.5 miles

Example 2: A ball is thrown upward with an initial velocity of 30 m/s. Its height after t seconds is given by h(t) = 30t - 5t. What is the maximum height reached by the ball?

Solution: To find the maximum height, we need to find the vertex of this parabola.
The t-coordinate of the vertex is: t = -b/(2a) = -30/(2(-5)) = -30/(-10) = 3 seconds
Substitute t = 3 into the function: h(3) = 30(3) - 5(3) = 90 - 45 = 45 meters
Therefore, the maximum height is 45 meters.

Example 3: Two trains start from stations 300 miles apart and travel toward each other. One train travels at 60 mph, and the other at 70 mph. How long will it take for the trains to meet?

Solution: Let t = time until the trains meet.
Distance traveled by first train: d = 60t
Distance traveled by second train: d = 70t
Since they meet when the sum of their distances equals 300 miles:
60t + 70t = 300
130t = 300
t = 300/130 = 30/13 2.31 hours
Therefore, the trains will meet in approximately 2.31 hours.

Cost Functions

Cost functions are mathematical models that describe the cost of production as a function of quantity. Understanding cost functions is essential in business, economics, and operations management.

Total Cost Function: C(x) = CF + V(x) (where CF is fixed cost and V(x) is variable cost)

Average Cost Function: AC(x) = C(x)/x

Profit Function: P(x) = R(x) - C(x) (where R(x) is revenue)

Example 1: A company has fixed costs of $5000 and variable costs of $25 per unit produced. Write the total cost function if x units are produced.

Solution: C(x) = 5000 + 25x

Example 2: Given the cost function C(x) = 5000 + 25x, find the average cost function and the average cost when 100 units are produced.

Solution: Average cost function: AC(x) = C(x)/x = (5000 + 25x)/x = 5000/x + 25
Average cost when 100 units are produced: AC(100) = 5000/100 + 25 = 50 + 25 = $75 per unit

Example 3: A company sells products at $40 each. With the cost function C(x) = 5000 + 25x, find the break-even point (where revenue equals cost).

Solution: Revenue function: R(x) = 40x
Set R(x) = C(x): 40x = 5000 + 25x
40x - 25x = 5000
15x = 5000
x = 5000/15 = 333.33
Therefore, the break-even point is approximately 333.33 units. Since we can't produce a fraction of a unit, the company needs to produce at least 334 units to break even.

Practice Problems

Problem 1: Given f(x) = x - 4x + 6x - 3, find f(-2).

Solution: f(-2) = (-2) - 4(-2) + 6(-2) - 3 = -8 - 4(4) - 12 - 3 = -8 - 16 - 12 - 3 = -39

Problem 2: Convert 70F to Celsius.

Solution: C = (5/9)(70 - 32) = (5/9)(38) 21.1C

Problem 3: If a car travels at 55 mph and travels 165 miles, how long did the journey take?

Solution: t = d/r = 165/55 = 3 hours

Problem 4: A ball is thrown upward with an initial velocity of 40 m/s. Its height after t seconds is given by h(t) = 40t - 5t. When does the ball return to the ground?

Solution: The ball returns to the ground when h(t) = 0:
40t - 5t = 0
5t(8 - t) = 0
t = 0 or t = 8
Since t = 0 corresponds to when the ball was thrown, the answer is t = 8 seconds.

Problem 5: A company has fixed costs of $8000 and variable costs of $15 per item. If each item sells for $30, find the profit function and determine the profit when 200 items are sold.

Solution: Cost function: C(x) = 8000 + 15x
Revenue function: R(x) = 30x
Profit function: P(x) = R(x) - C(x) = 30x - (8000 + 15x) = 15x - 8000
Profit when 200 items are sold: P(200) = 15(200) - 8000 = 3000 - 8000 = -$5000 (This is a loss, not a profit)

Further Study

Mastery of these precalculus topics provides a solid foundation for calculus. Students should continue practicing with increasingly complex problems. Additional areas of precalculus to review include:

  • Trigonometric functions and identities
  • Exponential and logarithmic models
  • Systems of equations
  • Sequences and series
  • Conic sections

Working through comprehensive textbooks and utilizing online resources can further enhance mathematical proficiency before embarking on calculus studies.

Reference Files For Precalculus Review Problems With Functions, Temperature Conversion, Motion, And Cost Functions
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