1) An antique dresser was purchased for $7500 in 2005. The dresser increases in value by 9% per year. Find the value of the dresser in 2025.
2) a) Find the slope of the secant to the curve f(x) = 0.25(4)^(x) - 2 between:
p1) x=0 and x = 1
p2) x=0 and x=0.5
p3) x=0 and x=0.1
p4) x=0 and x=0.01
b) Extend the result from part a) to determine the tangent to the curve at x = 0 accurate to 3 decimal places.
Thanks.
1. Value = 7500 + 7500*0.09*20 = $21,000
1) To find the value of the dresser in 2025, we need to calculate the compounded value of the initial purchase price over the years. The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (value of the dresser in 2025)
P = the principal amount (purchase price)
r = annual interest rate (as a decimal, so 9% becomes 0.09)
n = number of times interest is compounded per year (assuming it's compounded annually, so n = 1)
t = number of years
In this case, P = $7500, r = 0.09, n = 1, and t = 2025 - 2005 = 20.
Plugging these values into the formula:
A = 7500(1 + 0.09/1)^(1*20)
A = 7500(1.09)^20
A ≈ $22,465.33
Therefore, the value of the dresser in 2025 is approximately $22,465.33.
2) a) To find the slope of the secant to the curve f(x) = 0.25(4)^(x) - 2, we need to calculate the average rate of change between two given points.
For p1) x=0 and x=1:
Use the formula for slope:
slope = (f(x₂) - f(x₁)) / (x₂ - x₁)
slope = [0.25(4)^1 - 2 - 0.25(4)^0 - 2] / (1 - 0)
Simplifying,
slope = (1 - 2) / 1
slope = -1
For p2) x=0 and x=0.5:
slope = [0.25(4)^0.5 - 2 - 0.25(4)^0 - 2] / (0.5 - 0)
Simplifying,
slope = (0.5 - 2) / 0.5
slope = -3.5
For p3) x=0 and x=0.1:
slope = [0.25(4)^0.1 - 2 - 0.25(4)^0 - 2] / (0.1 - 0)
Simplifying,
slope = (0.549 - 2) / 0.1
slope ≈ -15.51
For p4) x=0 and x=0.01:
slope = [0.25(4)^0.01 - 2 - 0.25(4)^0 - 2] / (0.01 - 0)
Simplifying,
slope = (0.5174 - 2) / 0.01
slope ≈ -148.26
b) To find the tangent at x = 0, we extend the slope we obtained in part a) and use the point-slope form of the equation of a line:
y - y₁ = m(x - x₁)
For x₁ = 0 and m = -1, the equation becomes:
y - f(0) = -1(x - 0)
Simplifying,
y - (0.25(4)^0 - 2) = -x
y - (0.25 - 2) = -x
y - 1.75 = -x
Approximating to 3 decimal places, the tangent at x = 0 is:
y = -x + 1.750