Tasks 05-02 - The Derivative as Rate of Change

Section 05: Differential Calculus

Problem 1: Computing Derivatives from Definition (x)

Use the limit definition \(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\) to find the derivative at the given point:

\(f(x) = 5x - 3\) at \(a = 2\)
\(f(x) = x^2 + 2x\) at \(a = 1\)
\(f(x) = 4\) at \(a = 3\)
Explain what the derivative value tells you about each function’s behavior at the given point.

A ball is thrown upward. Its height (in meters) after \(t\) seconds is given by: \[h(t) = -5t^2 + 20t + 2\]

Compute the average rate of change of height from \(t = 1\) to \(t = 3\) seconds.
Estimate the instantaneous rate of change at \(t = 2\) by calculating \(\frac{h(2.01) - h(2)}{0.01}\).
Verify your answer from (b) by computing \(h'(2)\) using the limit definition.
Interpret the meaning of \(h'(2)\) in the context of the ball’s motion.

A manufacturer’s total cost function (in euros) for producing \(x\) units is: \[C(x) = 500 + 10x + 0.05x^2\]

Determine the average cost per unit when producing 100 units.
Compute \(C'(100)\) by calculating \(C(101) - C(100)\).
Investigate the behavior of marginal cost: Is it increasing, decreasing, or constant? Calculate \(C'(50)\) and \(C'(150)\) using the approximation method.
Argue whether the company should expand from 100 to 101 units if they can sell each unit for €25.

A company sells widgets. The demand function is \(p = 80 - 0.4x\), where \(p\) is the price per widget (in euros) and \(x\) is the quantity demanded.

Show that the revenue function is \(R(x) = 80x - 0.4x^2\).
Compute \(R'(75)\) using the approximation \(R(76) - R(75)\).
Assess whether the company should increase or decrease production if they are currently producing 75 widgets. Substantiate your answer.
Decide: At what production level is revenue maximized? (Hint: Find where \(R'(x) = 0\) by using the fact that \(R'(x) = 80 - 0.8x\).)

A small business has the following functions: - Cost: \(C(x) = 200 + 8x + 0.02x^2\) - Revenue: \(R(x) = 50x - 0.3x^2\)

where \(x\) is the number of items produced and sold.

Determine the profit function \(P(x) = R(x) - C(x)\).
Compute the marginal profit at \(x = 40\) using \(P(41) - P(40)\).
Give the production level that maximizes profit using the formula \(P'(x) = R'(x) - C'(x)\) and the fact that:
- \(R'(x) = 50 - 0.6x\)
- \(C'(x) = 8 + 0.04x\)
Verify that this production level gives \(MR = MC\).
Graph the profit function and mark the optimal production point.

Use the limit definition to find \(f'(x)\) as a function of \(x\) (not just at a specific point) for \(f(x) = x^2 + 3x\).
Show that your answer from (a) satisfies the property that \(f'(2) = 7\) by direct evaluation.
Investigate: For what value of \(x\) is the tangent line to \(f\) horizontal? Give the point on the curve where this occurs.
Verify by computing \(f'(1)\) and finding the equation of the tangent line at \((1, f(1))\).

A tech startup develops mobile apps. Their analysis shows:

where \(x\) is the number of apps developed per month.

Determine the profit function \(P(x)\).
Compute the average cost per app when developing 20 apps per month.
Show that the marginal cost and marginal revenue at \(x = 20\) are:
- \(MC(20) \approx 1200\) euros/app
- \(MR(20) \approx 1400\) euros/app

Decide whether the company should increase or decrease production from 20 apps. Substantiate mathematically.
Investigate the optimal production level by finding where \(P'(x) = 0\), using:
- \(R'(x) = 2000 - 30x\)
- \(C'(x) = 800 + 40x\)
Give the maximum profit and verify that \(MR = MC\) at this production level.

Argue whether it’s worth developing apps if the startup can only manage 10 apps per month.
Assess the break-even points (where \(P(x) = 0\)) and interpret their business significance.