Topics in this course

1.Limits and Continuity
A limit describes the value a function approaches as its input approaches some number, whether or not the function is defined there. Writing $\lim_{x \to a} f(x) = L$ means $f(x)$ can be made as close to $L$ as you like by taking $x$ close enough to $a$ , from both sides.
- If $f$ is continuous at $a$ , the limit is simply $f(a)$ .
- A two-sided limit exists only if the left and right limits agree.
- $\lim_{x \to 0} \frac{\sin x}{x} = 1$ with $x$ in radians.
Quick check: Evaluate $\lim_{x \to 2} (3x^2 - 1)$ .
$11$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study limits and continuity
2.Derivatives
The derivative $f'(x)$ is the instantaneous rate of change of $f$ at $x$ , and geometrically it is the slope of the tangent line. It is defined as the limit $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ , but in practice you rarely use the definition, because a handful of rules cover almost every function.
- Power rule: $\frac{d}{dx} x^n = n x^{n-1}$ .
- Product rule: $(uv)' = u'v + uv'$ ; quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ .
- Chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$ .
Quick check: Find $f'(2)$ for $f(x) = x^3 - 4x$ .
$8$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study derivatives
3.Applications of Derivatives
Because the derivative measures slope, its sign tells you where a function rises ( $f' > 0$ ) or falls ( $f' < 0$ ). Points where $f' = 0$ or $f'$ is undefined are critical points, the only candidates for local maxima and minima. The second derivative describes concavity and confirms which candidates are peaks and which are valleys.
- First derivative test: a sign change from $+$ to $-$ marks a local maximum; $-$ to $+$ a local minimum.
- Second derivative test: $f''(c) < 0$ at a critical point means a maximum, $f''(c) > 0$ a minimum.
- On a closed interval, compare the function at critical points and at both endpoints.
Quick check: Find the $x$ -coordinate of the minimum of $f(x) = x^2 - 6x + 2$ .
$3$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study applications of derivatives

Practice tests

Derivatives Practice Test · Eight questions on the derivative rules and their meaning.

Calculus 1 course

Topics in this course

1.Limits and Continuity

2.Derivatives

3.Applications of Derivatives

Practice tests

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