1.3) exponential function.
An exponential function is a type of function in which the independent variable appears as an exponent. The general form of an exponential function is:
f(x) = a^x
where a is a constant and x is the independent variable. The value of a determines the growth rate or decay rate of the function.
Exponential functions are widely used in various fields, such as finance, economics, biology, physics, and engineering. Some examples of exponential phenomena are population growth, compound interest, radioactive decay, and bacterial growth.
Here are some sample problems involving exponential functions:
Problem 1: Write the exponential function y = 3(2)^x in the form y = a(b)^x.
Solution: We can write 2 as (2/3)^(-1), so:
y = 3(2)^x y = 3[(2/3)^(-1)]^x y = 3(2/3)^(-x) y = 3(3/2)^x
Therefore, the function can be written as y = 3(3/2)^x, which is in the form y = a(b)^x.
Problem 2: A certain species of bacteria doubles every 3 hours. If the initial population is 1000 bacteria, find the population after 6 hours.
Solution: We can model the population using the exponential function P(t) = 1000(2)^(t/3), where t is the time in hours. To find the population after 6 hours, we substitute t = 6 into the function: P(6) = 1000(2)^(6/3) = 1000(2)^2 = 4000. Therefore, the population after 6 hours is 4000 bacteria.
Problem 3: A car is purchased for $20,000 and is expected to depreciate by 15% per year. Write an equation to model the value V of the car after t years, and find the value of the car after 4 years.
Solution: The value of the car after t years can be modeled using the exponential function V(t) = 20000(0.85)^t, where t is the time in years. To find the value of the car after 4 years, we substitute t = 4 into the function: V(4) = 20000(0.85)^4 ≈ 9972. Therefore, the value of the car after 4 years is approximately $9,972.
1.4) Trigonometric functions
Trigonometric functions are a set of functions that relate angles to the ratios of the sides of a right triangle. The six basic trigonometric functions are:
- sine (sin)
- cosine (cos)
- tangent (tan)
- cosecant (csc)
- secant (sec)
- cotangent (cot)
These functions are defined based on the ratios of the sides of a right triangle as follows:
sinθ = opposite/hypotenuse cosθ = adjacent/hypotenuse tanθ = opposite/adjacent cscθ = hypotenuse/opposite secθ = hypotenuse/adjacent cotθ = adjacent/opposite
where θ is the angle in radians or degrees.
Trigonometric functions have many applications in mathematics, physics, engineering, and other fields. They are used to solve problems involving triangles, waves, periodic phenomena, and oscillations, among others.
Here are some sample problems involving trigonometric functions:
Problem 1: Find the value of sin 60°.
Solution: From the definition of sine, we have:
sin 60° = opposite/hypotenuse
For a 30-60-90 triangle with hypotenuse 1, the opposite side is √3/2. Therefore:
sin 60° = √3/2
Problem 2: Find the value of cos π/3.
Solution: From the definition of cosine, we have:
cos π/3 = adjacent/hypotenuse
For a 30-60-90 triangle with hypotenuse 1, the adjacent side is 1/2. Therefore:
cos π/3 = 1/2
Problem 3: Find the value of tan 45°.
Solution: From the definition of tangent, we have:
tan 45° = opposite/adjacent
For a 45-45-90 triangle with hypotenuse 1, the opposite side and adjacent side are equal, so:
tan 45° = 1
Problem 4: Find the value of sec 30°.
Solution: From the definition of secant, we have:
sec 30° = hypotenuse/adjacent
For a 30-60-90 triangle with hypotenuse 1, the adjacent side is √3/2. Therefore:
sec 30° = 2/√3
Problem 5: Find the value of cot π/6.
Solution: From the definition of cotangent, we have:
cot π/6 = adjacent/opposite
For a 30-60-90 triangle with hypotenuse 1, the adjacent side is √3/2 and the opposite side is 1/2. Therefore:
cot π/6 = (√3/2)/(1/2) = √3
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Post added on Mar 27, 2023 at 12:26 PM
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