Applications of Derivatives

The derivative works like a compass that tells you whether a function's graph is currently climbing uphill or sliding downhill. A positive slope means it is rising, a negative slope means it is falling, and the exact spot where the sign flips is a peak (local max) or a valley (local min). Once you grasp this, a single derivative lets you read off a function's maximum and minimum values, and even the velocity and acceleration of a moving object. Drag the x-position slider and watch how the tangent tips over and how the green (increasing) and red (decreasing) regions trade places.

Increasing vs Decreasing

📈 What the Sign of f' Tells You

①f'(x) > 0 ⇒ f(x) increasing

②f'(x) < 0 ⇒ f(x) decreasing

③f'(x) = 0 ⇒ candidate for local max/min

Extrema & Sign Chart

x position0

Extremum Test

f'(a) = 0 with sign + → - ⇒ local max, - → + ⇒ local min

Use a sign chart for the derivative

💡 Building a Sign Chart

①Solve f'(x) = 0 (e.g., 3x² − 3 = 0 → x = ±1)

②Determine the sign of f'(x) on each interval

③Sign change marks a local extremum

④No sign change at a critical point ⇒ NOT an extremum (e.g., x³ at x = 0)

Maxima & Minima

Max/Min on a Closed Interval

[a, b] candidates: ①f'(x)=0 points ②endpoints f(a), f(b)

Compare candidate values; max wins, min wins

Second Derivative Test

f'(a)=0, f''(a)<0 ⇒ local max / f'(a)=0, f''(a)>0 ⇒ local min

Decide max/min by the sign of f''

🔑 Max-Min Strategy

①Closed interval: compare extrema + endpoints

②Open interval: extrema alone

③Real-world optimization: set up an objective and differentiate

Velocity & Acceleration

Time t1 s

Position-Velocity-Acceleration

v(t) = s'(t), a(t) = v'(t) = s''(t)

Position → velocity → acceleration

💡 Reading Motion

①v(t) > 0: moving in + direction / v(t) < 0: − direction

②v(t) = 0: turning point (peak)

③Constant a(t) ⇒ uniformly accelerated motion (e.g., free fall)

Wrap-up

Extremum Condition

f'(a) = 0 with sign change ⇒ local max/min

Sign change of f' is what matters

🎯 Exam Points

①Sign chart: solve f'(x) = 0, then check signs on each interval

②Test: f' + → − is max; − → + is min

③Closed interval: compare extrema + endpoints

④Tangent line: y = f(a) + f'(a)(x − a)

⑤Use the sign chart to sketch and solve equations/inequalities

Worked Examples & Past Exam

Example 1

Find the local maximum and minimum of f(x) = x³ − 3x.

Find the derivative and solve f'(x) = 0.

f'(x) = 3x² - 3 = 0 ⟹ x = ±1

By the sign of f', x = −1 gives a maximum and x = 1 a minimum.

local max f(-1) = 2, local min f(1) = -2

▸ local max 2, local min −2

Where f'(x) = 0, a sign change +→− gives a max and −→+ gives a min.

Example 2

A point P starts at the origin with position s(t) = t² − 4t. Find the velocity v(1) at t = 1.

Velocity is the derivative of position. v(t) = s'(t).

v(t) = s'(t) = 2t - 4

Substitute t = 1.

v(1) = 2(1) - 4 = -2

▸ v(1) = −2

Differentiating position once gives velocity; its sign shows the direction of motion.

2022 KICE mock exam Math type, adapted

What is the slope of the tangent to y = x² + 1 at the point (1, 2)?

①2

②1

③3

④4

⑤0

▸ ① 2

The tangent slope is the derivative at that point. y' = 2x.

y' = (x² + 1)' = 2x

Substitute x = 1.

y'(1) = 2(1) = 2

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Derivative & Differentiation

Indefinite Integral

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