Determine f'(x) from first principles if f(x) = x³

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Here's the solution to Question 7 and Question 8.

QUESTION 7

7.1 Calculate the scrap value of the server at the end of 6 years.
The value depreciates using the diminishing-balance method.

Use the diminishing-balance depreciation formula $A = P(1 - i)^n$ .
Given: $P = \text{R}500\,000$ , $i = 20\% = 0.20$ , $n = 6$ years.
$A = 500\,000(1 - 0.20)^6$ $A = 500\,000(0.8)^6$ $A = 500\,000(0.262144)$ $A = 131\,072$
The scrap value of the server is $\boxed{\text{R}131\,072.00}$ .

7.2 Calculate the cost of the new server if the inflation rate is at 7% per annum.
Use the compound interest formula for inflation $A = P(1 + i)^n$ .
Given: $P = \text{R}500\,000$ , $i = 7\% = 0.07$ , $n = 6$ years.
$A = 500\,000(1 + 0.07)^6$ $A = 500\,000(1.07)^6$ $A = 500\,000(1.50073035)$ $A = 750\,365.175$
The cost of the new server is $\boxed{\text{R}750\,365.18}$ .

7.3.1 Calculate the value of the monthly installment into the sinking fund.
The sinking fund needs to accumulate the cost of the new server (from 7.2), which is $\text{R}750\,365.18$ .
Interest is 8% per annum compounded monthly. Payments are made immediately, and the last payment is at the end of the 6-year period. This implies an annuity due with 72 payments.
Identify the variables.
$FV = \text{R}750\,365.18$
$i = \frac{0.08}{12}$ per month
$n = 6 \text{ years} \times 12 \text{ months/year} = 72 \text{ payments}$
Use the future value of an annuity due formula
$FV = PMT \frac{(1+i)^n - 1}{i} (1+i)$ .
$750\,365.18 = PMT \frac{\left(1 + \frac{0.08}{12}\right)^{72} - 1}{\frac{0.08}{12}} \left(1 + \frac{0.08}{12}\right)$
Calculate the factor.
$\frac{\left(1 + \frac{0.08}{12}\right)^{72} - 1}{\frac{0.08}{12}} \left(1 + \frac{0.08}{12}\right) \approx \frac{(1.006666667)^{72} - 1}{0.006666667} (1.006666667)$ $\approx \frac{1.61303639 - 1}{0.006666667} (1.006666667)$ $\approx \frac{0.61303639}{0.006666667} (1.006666667)$ $\approx 91.9554585 \times 1.006666667 \approx 92.56914$
Solve for $PMT$ .
$PMT = \frac{750\,365.18}{92.56914}$ $PMT \approx 8106.00$
The monthly installment into the sinking fund is $\boxed{\text{R}8106.00}$ .

7.4 After how many months will there be more than R1 000 000 in the fund?
The business pays a monthly installment of R15 000. Assuming payments are made at the beginning of each month (annuity due, consistent with 7.3.1).
Identify the variables.
$FV = \text{R}1\,000\,000$
$PMT = \text{R}15\,000$
$i = \frac{0.08}{12}$ per month
Use the future value of an annuity due formula
$FV = PMT \frac{(1+i)^n - 1}{i} (1+i)$ .
$1\,000\,000 = 15\,000 \frac{\left(1 + \frac{0.08}{12}\right)^n - 1}{\frac{0.08}{12}} \left(1 + \frac{0.08}{12}\right)$
Isolate the term with $n$ .
$\frac{1\,000\,000 \times \frac{0.08}{12}}{15\,000 \times \left(1 + \frac{0.08}{12}\right)} = \left(1 + \frac{0.08}{12}\right)^n - 1$ $\frac{6666.66667}{15\,000 \times 1.006666667} = \left(1 + \frac{0.08}{12}\right)^n - 1$ $\frac{6666.66667}{15100} = \left(1 + \frac{0.08}{12}\right)^n - 1$ $0.441501 = \left(1 + \frac{0.08}{12}\right)^n - 1$ $1.441501 = \left(1 + \frac{0.08}{12}\right)^n$
Solve for $n$ using logarithms.
$\ln(1.441501) = n \ln\left(1 + \frac{0.08}{12}\right)$ $\ln(1.441501) = n \ln(1.006666667)$ $0.36569 \approx n \times 0.0066444$ $n \approx \frac{0.36569}{0.0066444} \approx 55.04$
Since the question asks "more than R1 000 000", we round up to the next whole month.
After $\boxed{56 \text{ months}}$ there will be more than R1 000 000 in the fund.

QUESTION 8

8.1 Determine $f'(x)$ from first principles if $f(x) = x^3$ .
Write down the definition of the derivative from first principles.
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Substitute $f(x) = x^3$ and $f(x+h) = (x+h)^3$ into the formula.
Recall $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$ .
$f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h}$
Simplify the numerator.
$f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h}$
Factor out $h$ from the numerator and cancel it with the denominator.
$f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2)$
Apply the limit as $h \to 0$ .
$f'(x) = 3x^2 + 3x(0) + (0)^2$ $f'(x) = \boxed{3x^2}$
8.2 Determine:

8.2.1 $f'(x)$ if $f(x) = \left(x + \frac{1}{x}\right)^2$ .
Expand the function $f(x)$ .
$f(x) = \left(x + x^{-1}\right)^2$ $f(x) = x^2 + 2(x)(x^{-1}) + (x^{-1})^2$ $f(x) = x^2 + 2 + x^{-2}$
Differentiate $f(x)$ term by term.
$f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(2) + \frac{d}{dx}(x^{-2})$ $f'(x) = 2x + 0 - 2x^{-3}$ $f'(x) = \boxed{2x - \frac{2}{x^3}}$
8.2.2 $\frac{dy}{dx}$ if $y = \frac{3}{x^3} - \pi x$ .
Rewrite the function using negative exponents.
$y = 3x^{-3} - \pi x$
Differentiate $y$ term by term with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(3x^{-3}) - \frac{d}{dx}(\pi x)$ $\frac{dy}{dx} = 3(-3)x^{-3-1} - \pi(1)x^{1-1}$ $\frac{dy}{dx} = -9x^{-4} - \pi$ $\frac{dy}{dx} = \boxed{-\frac{9}{x^4} - \pi}$

Determine f'(x) from first principles if f(x) = x³

Step-by-step solution

Use the diminishing-balance depreciation formula $A = P(1 - i)^n$ .

Use the compound interest formula for inflation $A = P(1 + i)^n$ .

Identify the variables.

Use the future value of an annuity due formula

Calculate the factor.

Solve for $PMT$ .

Identify the variables.

Use the future value of an annuity due formula

Isolate the term with $n$ .

Solve for $n$ using logarithms.

Write down the definition of the derivative from first principles.

Substitute $f(x) = x^3$ and $f(x+h) = (x+h)^3$ into the formula.

Simplify the numerator.

Factor out $h$ from the numerator and cancel it with the denominator.

Apply the limit as $h \to 0$ .

Expand the function $f(x)$ .

Differentiate $f(x)$ term by term.

Rewrite the function using negative exponents.

Differentiate $y$ term by term with respect to $x$ .

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More Mathematics Questions

Determine f'(x) from first principles if f(x) = x³

Step-by-step solution

Use the diminishing-balance depreciation formula A=P(1−i)nA = P(1 - i)^nA=P(1−i)n.

Use the compound interest formula for inflation A=P(1+i)nA = P(1 + i)^nA=P(1+i)n.

Identify the variables.

Use the future value of an annuity due formula

Calculate the factor.

Solve for PMTPMTPMT.

Identify the variables.

Use the future value of an annuity due formula

Isolate the term with nnn.

Solve for nnn using logarithms.

Write down the definition of the derivative from first principles.

Substitute f(x)=x3f(x) = x^3f(x)=x3 and f(x+h)=(x+h)3f(x+h) = (x+h)^3f(x+h)=(x+h)3 into the formula.

Simplify the numerator.

Factor out hhh from the numerator and cancel it with the denominator.

Apply the limit as h→0h \to 0h→0.

Expand the function f(x)f(x)f(x).

Differentiate f(x)f(x)f(x) term by term.

Rewrite the function using negative exponents.

Differentiate yyy term by term with respect to xxx.

Still stuck on this one?

More Mathematics Questions

Use the diminishing-balance depreciation formula $A = P(1 - i)^n$ .

Use the compound interest formula for inflation $A = P(1 + i)^n$ .

Solve for $PMT$ .

Isolate the term with $n$ .

Solve for $n$ using logarithms.

Substitute $f(x) = x^3$ and $f(x+h) = (x+h)^3$ into the formula.

Factor out $h$ from the numerator and cancel it with the denominator.

Apply the limit as $h \to 0$ .

Expand the function $f(x)$ .

Differentiate $f(x)$ term by term.

Differentiate $y$ term by term with respect to $x$ .