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Applications of Differentiation

$f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$

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Rates of Change (Differentiation)

Function $f(x)$ (e.g., x^3 - 2*x + 1):

Point $x_0$ for slope calculation:

Enter function and point, then calculate.

Velocity and Acceleration

Position function $s(t)$ (e.g., t^3 - 6*t^2 + 9*t):

Time $t_0$ for calculation:

Position $s(t)$

Velocity $v(t)$

Acceleration $a(t)$

Understanding Turning Points

A turning point (or stationary point) on the graph of a function $y=f(x)$ is a point where the derivative $f'(x)$ is zero or undefined. At these points, the tangent to the curve is horizontal (if $f'(x)=0$).

Turning points can be classified as:

Local Maximum: The function value is greater than or equal to the values at nearby points. The curve changes from increasing to decreasing.
Local Minimum: The function value is less than or equal to the values at nearby points. The curve changes from decreasing to increasing.
Stationary Point of Inflexion: A point where $f'(x)=0$ but it's neither a local maximum nor a local minimum (e.g., $y=x^3$ at $x=0$).

Tests for Nature of Turning Points (where $f'(c)=0$):

First Derivative Test:

If $f'(x)$ changes from positive to negative at $x=c$, then $f(c)$ is a local maximum.
If $f'(x)$ changes from negative to positive at $x=c$, then $f(c)$ is a local minimum.
If $f'(x)$ does not change sign at $x=c$, then $f(c)$ is a stationary point of inflexion.

Second Derivative Test:

If $f''(c) < 0$, then $f(c)$ is a local maximum.
If $f''(c) > 0$, then $f(c)$ is a local minimum.
If $f''(c) = 0$, the test is inconclusive; use the first derivative test.

Interactive Differentiation Practice

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Determine Turning Points and Their Nature

Function $f(x)$ (e.g., x^3 - 3*x^2 + 2):

Plotting Interval Start $x_{min}$ (optional):

Plotting Interval End $x_{max}$ (optional):

Note: Finding roots for $f'(x)=0$ for complex functions can be challenging. This solver attempts to find them for polynomials and some common functions. For more complex cases, results may be approximate or incomplete.

Practical Problems Involving Maxima and Minima

Function $f(x)$:

Interval Start $a$:

Interval End $b$:

Points of Inflexion

Function $f(x)$ (e.g., x^4 - 4*x^3):

Plotting Interval Start $x_{min}$ (optional):

Plotting Interval End $x_{max}$ (optional):

Note: Finding roots for $f''(x)=0$ can be challenging. This solver attempts to find them for polynomials and some common functions.

Tangents and Normals to a Curve

Function $f(x)$ (e.g., x^2):

Point $x_0$ for tangent/normal:

Small Changes and Approximations

Function $f(x)$ (e.g., sqrt(x)):

Initial point $x$:

Small change $\Delta x$ (or $dx$):

Differentiation of Parametric Equations

Function $x(t)$ (e.g., 2*t):

Function $y(t)$ (e.g., t^2 - 1):

Value of $t$ for calculation (e.g., 1):

Plotting Interval $t_{min}$ (optional):

Plotting Interval $t_{max}$ (optional):