Calculus 1 : How to find trapezoidal approximation by graphing functions

Example Questions

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Example Question #1 : How To Find Trapezoidal Approximation By Graphing Functions

Use the trapezoidal approximation to find the area under the curve using the graph with four partitions.

Explanation:

The trapezoid rule states that

.

Therefore, using our graph, we have:

We find the function values at the sample point:

Then we substitute the appropriate values into the trapezoid rule approximation:

Example Question #2 : Trapezoidal Approximation

Is the following function increasing or decreasing at the point ?

h(x) is increasing at , because the first derivative is positive.

h(x) is increasing at , because the second derivative is positive.

h(x) is decreasing at , because the first derivative is negative.

h(x) is decreasing at , because the second derivative is negative.

h(x) is decreasing at , because the first derivative is negative.

Explanation:

Is the following function increasing or decreasing at the point ?

Increasing and decreasing intervals can be found via the first derivative. Since derivatives measure rates of change, the sign of the derivative at a given point can tell you whether a function is increasing or decreasing.

Begin by taking the derivative of our function:

Becomes:

Next, find h'(-6) and look at the sign.

So, our first derivative is very negative at the given point. This means that h(x) is decreasing.

Example Question #3 : Trapezoidal Approximation

Use the trapezoidal approximation to approximate the following integral:

Explanation:

The trapezoidal approximation of a definite integral is given by the following formula:

Using the above formula, we get

Example Question #4 : Trapezoidal Approximation

Use the trapezoidal approximation to evaluate the following integral:

Explanation:

To evaluate a definite integral using the trapezoidal approximation, we must use the formula

Using the above formula, we get

Example Question #5 : Trapezoidal Approximation

Use the trapezoidal approximation to evaluate the following integral:

Explanation:

To evaluate a definite integral using the trapezoidal approximation, we must use the formula

Using the above formula, we get

Example Question #6 : Trapezoidal Approximation

Evaluate the following integral using the trapezoidal approximation:

Explanation:

To evaluate the integral using the trapezoidal rule, we must use the formula

Using the above formula, we get the following:

Example Question #7 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

Explanation:

To evaluate a definite integral using the trapezoidal approximation, we must use the following formula:

So, using the above formula, we get

which simplifies to

Example Question #8 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

Explanation:

To evaluate the definite integral using the trapezoidal approximation, we must use the following formula:

Using the above formula, we get

Example Question #9 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

Explanation:

To evaluate the integral using the trapezoidal approximation, we must use the following formula:

Using the formula, we get

Example Question #10 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation: