Unit 3: Calculus (Part 1)

Differentiation & Applications [Syllabus 3.1 – 3.5]

1. Derivatives up to Second Order

Parametric Functions If $$x = f(t)$$and$$y = g(t)$$, then:
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$$(Provided$$f'(t) \neq 0$$)
Implicit Functions When $$y$$cannot be expressed explicitly as$$y=f(x)$$(e.g.,$$x^2 + y^2 = a^2$$).
Method: Differentiate both sides w.r.t $$x$$and group terms involving$$\frac{dy}{dx}$$.
Logarithmic Differentiation Used for functions like $$y = [f(x)]^{g(x)}$$ or product of many functions.
Step 1: Take $$\log$$ on both sides.
Step 2: Differentiate w.r.t $$x$$.
Second Order Derivative If $$y = f(x)$$, then $$\frac{dy}{dx} = f'(x)$$ (First Order)
$$\frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d^2y}{dx^2} = f”(x)$$ (Second Order)

2. Rate of Change & Marginal Cost/Revenue

Rate of Change $$\frac{dy}{dx}$$represents the rate of change of$$y$$with respect to$$x$$.
Value at $$x=x_0$$is denoted by$$\left(\frac{dy}{dx}\right)_{x=x_0}$$
Marginal Cost (MC) Instantaneous rate of change of Total Cost $$C(x)$$w.r.t output$$x$$.
$$MC = \frac{d}{dx}(C(x))$$
Marginal Revenue (MR) Instantaneous rate of change of Total Revenue $$R(x)$$w.r.t sales$$x$$.
$$MR = \frac{d}{dx}(R(x))$$
Revenue Function $$R(x) = p \times x$$
(where $$p$$is price per unit and$$x$$ is number of units)

3. Increasing and Decreasing Functions

Let $$f$$be continuous on$$[a, b]$$and differentiable on$$(a, b)$$.

Strictly Increasing $$f'(x) > 0$$for all$$x \in (a, b)$$
Increasing $$f'(x) \ge 0$$for all$$x \in (a, b)$$
Strictly Decreasing $$f'(x) < 0$$for all$$x \in (a, b)$$
Decreasing $$f'(x) \le 0$$for all$$x \in (a, b)$$

4. Maxima and Minima

Critical Points Points where $$f'(x) = 0$$or$$f'(x)$$ is not defined.
Second Derivative Test Find $$c$$such that$$f'(c)=0$$. Then find $$f”(c)$$:
1. If $$f”(c) < 0 \Rightarrow$$Local Maxima at$$c$$.
2. If $$f”(c) > 0 \Rightarrow$$Local Minima at$$c$$.
3. If $$f”(c) = 0 \Rightarrow$$ Test Fails (Use First Derivative Test).
Absolute Max/Min To find absolute max/min in closed interval $$[a, b]$$:
1. Find critical points inside $$(a, b)$$.
2. Evaluate $$f(x)$$at critical points AND endpoints$$a, b$$.
3. Identify the largest and smallest values.
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