Explain the core concept of the Fundamental Theorem of Calculus.

It establishes the relationship between differentiation and integration, showing that they are inverse operations.

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Explain the core concept of the Fundamental Theorem of Calculus.

It establishes the relationship between differentiation and integration, showing that they are inverse operations.

Explain how the Fundamental Theorem of Calculus is used to find derivatives of integrals.

It states that the derivative of an integral with a variable upper limit is simply the integrand evaluated at that upper limit.

What happens when the upper bound of the integral is a function of x?

You must apply the chain rule by multiplying the integrand evaluated at that function by the derivative of the function.

What is the significance of the Fundamental Theorem of Calculus?

It provides a method to evaluate definite integrals and connects the concepts of area and rate of change.

How do you handle a constant of integration when using the Fundamental Theorem of Calculus?

For definite integrals, the constant of integration cancels out when evaluating at the upper and lower limits.

Explain the relationship between the area under a curve and the antiderivative.

The definite integral, calculated using the antiderivative, gives the net area under the curve between the limits of integration.

Why is continuity important for the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus requires the function to be continuous on the interval to ensure the existence of the integral and derivative.

What does the Fundamental Theorem of Calculus tell us about displacement and velocity?

The integral of velocity gives displacement, showing how integration can be used to find position from velocity.

Explain how the Fundamental Theorem of Calculus simplifies finding areas.

It provides a direct method to calculate the area under a curve by finding the antiderivative and evaluating at the bounds.

What are some common mistakes when applying the Fundamental Theorem of Calculus?

Forgetting the chain rule when the upper limit is a function, or not switching the bounds when needed.

Define Accumulation Function.

A function defined by a definite integral where one of the limits of integration is a variable.

What does the Fundamental Theorem of Calculus relate?

Differentiation and integration; it shows they are inverse processes.

What is the integrand?

The function inside the integral that is being integrated.

What is a definite integral?

An integral with upper and lower limits, resulting in a numerical value.

What is an antiderivative?

A function whose derivative is the original function.

What is the upper bound of an integral?

The upper limit of integration in a definite integral.

What is the lower bound of an integral?

The lower limit of integration in a definite integral.

Define the Chain Rule.

A rule for differentiating composite functions.

What is the derivative?

The instantaneous rate of change of a function with respect to its variable.

What is integration?

The process of finding the area under a curve.

How to find $g'(x)$ if $g(x) = \int_{a}^{x} f(t) dt$ ?

Apply the Fundamental Theorem of Calculus: $g'(x) = f(x)$ .

How to find $g'(x)$ if $g(x) = \int_{a}^{u(x)} f(t) dt$ ?

Apply the Fundamental Theorem of Calculus and the Chain Rule: $g'(x) = f(u(x)) cdot u'(x)$ .

How to find $g'(c)$ given $g(x) = \int_{a}^{x} f(t) dt$ ?

First find $g'(x)$ using the Fundamental Theorem of Calculus, then evaluate $g'(c)$ .

How to find $F'(x)$ if $F(x) = \int_{u(x)}^{b} f(t) dt$ ?

First, switch the limits: $F(x) = -\int_{b}^{u(x)} f(t) dt$ . Then, $F'(x) = -f(u(x)) cdot u'(x)$ .

How to find $g'(x)$ if $g(x) = \int_{u(x)}^{v(x)} f(t) dt$ ?

Split the integral: $g(x) = \int_{0}^{v(x)} f(t) dt - \int_{0}^{u(x)} f(t) dt$ . Then, $g'(x) = f(v(x))v'(x) - f(u(x))u'(x)$ .

How do you evaluate $\frac{d}{dx} \int_{2}^{x^3} \cos(t) dt$ ?

Apply the Fundamental Theorem of Calculus and the chain rule: $\cos(x^3) cdot 3x^2$ .

How do you find $F'(x)$ if $F(x) = \int_{5}^{x} (t^2 + 1) dt$ ?

Apply the Fundamental Theorem of Calculus: $F'(x) = x^2 + 1$ .

How do you solve for $g'(2)$ if $g(x) = \int_{1}^{x^2} \sqrt{t} dt$ ?

First find $g'(x) = \sqrt{x^2} cdot 2x = |x| cdot 2x$ . Then, $g'(2) = \sqrt{2^2} cdot 2(2) = 8$ .

How do you find the derivative of $\int_{x}^{0} e^{t^2} dt$ ?

Switch the bounds and apply the Fundamental Theorem of Calculus: $-\int_{0}^{x} e^{t^2} dt$ , so the derivative is $-e^{x^2}$ .

How do you find $h'(x)$ if $h(x) = \int_{\sin(x)}^{3} t^3 dt$ ?

Switch the bounds: $h(x) = -\int_{3}^{\sin(x)} t^3 dt$ . Apply the Fundamental Theorem of Calculus and the chain rule: $h'(x) = -(\sin(x))^3 \cos(x)$ .