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Calculus

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What is a limit in calculus?

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A limit describes the value that a function approaches as the input gets closer to a specific value, written as $\lim_{x \to a} f(x) = L$ .

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What is a limit in calculus?

A limit describes the value that a function approaches as the input gets closer to a specific value, written as $$\lim_{x \to a} f(x) = L$$.

What does the derivative of a function represent?

The derivative represents the rate of change of a function, defined as $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$.

What does it mean if $$f'(x) > 0$$?

It means that the function $$f(x)$$ is increasing on that interval.

What does it mean if $$f'(x) < 0$$?

It means that the function $$f(x)$$ is decreasing on that interval.

What does it mean if $$f'(x) = 0$$ at a point?

That point could be a local maximum, minimum, or inflection point.

What is the power rule for derivatives?

For $$f(x) = x^n$$, the derivative is $$f'(x) = nx^{n-1}$$.

What is the equation of the tangent line at point (a, f(a))?

The tangent line equation is $$y - f(a) = f'(a)(x - a)$$.

What is the equation of the normal line at point (a, f(a))?

The normal line equation is $$y - f(a) = -\frac{1}{f'(a)}(x - a)$$.

What is the indefinite integral of a function?

The indefinite integral is written as $$\int f(x) dx = F(x) + C$$, where $$F(x)$$ is an antiderivative and $$C$$ is a constant.

What is the derivative of $$\sin x$$?

The derivative is $$\frac{d}{dx}(\sin x) = \cos x$$.

What is the derivative of $$e^x$$?

The derivative is $$\frac{d}{dx}(e^x) = e^x$$.

What is the Chain Rule?

The Chain Rule states that $$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$.

What is the Product Rule?

The Product Rule states that $$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$$.

What is the Quotient Rule?

The Quotient Rule states that $$(\frac{f(x)}{g(x)})' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$$.

What does the second derivative indicate about a function?

The second derivative $$f''(x)$$ indicates concavity: if $$f''(x) > 0$$, the function is concave up; if $$f''(x) < 0$$, it is concave down.

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1 / 15

What is a limit in calculus?

Flip

Spacebar

A limit describes the value that a function approaches as the input gets closer to a specific value, written as $\lim_{x \to a} f(x) = L$ .

Flip

Spacebar

What is a limit in calculus?

A limit describes the value that a function approaches as the input gets closer to a specific value, written as $$\lim_{x \to a} f(x) = L$$.

What does the derivative of a function represent?

The derivative represents the rate of change of a function, defined as $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$.

What does it mean if $$f'(x) > 0$$?

It means that the function $$f(x)$$ is increasing on that interval.

What does it mean if $$f'(x) < 0$$?

It means that the function $$f(x)$$ is decreasing on that interval.

What does it mean if $$f'(x) = 0$$ at a point?

That point could be a local maximum, minimum, or inflection point.

What is the power rule for derivatives?

For $$f(x) = x^n$$, the derivative is $$f'(x) = nx^{n-1}$$.

What is the equation of the tangent line at point (a, f(a))?

The tangent line equation is $$y - f(a) = f'(a)(x - a)$$.

What is the equation of the normal line at point (a, f(a))?

The normal line equation is $$y - f(a) = -\frac{1}{f'(a)}(x - a)$$.

What is the indefinite integral of a function?

The indefinite integral is written as $$\int f(x) dx = F(x) + C$$, where $$F(x)$$ is an antiderivative and $$C$$ is a constant.

What is the derivative of $$\sin x$$?

The derivative is $$\frac{d}{dx}(\sin x) = \cos x$$.

What is the derivative of $$e^x$$?

The derivative is $$\frac{d}{dx}(e^x) = e^x$$.

What is the Chain Rule?

The Chain Rule states that $$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$.

What is the Product Rule?

The Product Rule states that $$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$$.

What is the Quotient Rule?

The Quotient Rule states that $$(\frac{f(x)}{g(x)})' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$$.

What does the second derivative indicate about a function?

The second derivative $$f''(x)$$ indicates concavity: if $$f''(x) > 0$$, the function is concave up; if $$f''(x) < 0$$, it is concave down.