All Calculus 1 Resources
Example Questions
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.
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 : How To Find Trapezoidal Approximation By Graphing Functions
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 second derivative is negative.
h(x) is decreasing at , because the first derivative is negative.
h(x) is decreasing at , because the first derivative is negative.
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 #1 : Trapezoidal Approximation
Use the trapezoidal approximation to approximate the following integral:
The trapezoidal approximation of a definite integral is given by the following formula:
Using the above formula, we get
Example Question #2 : How To Find Trapezoidal Approximation By Graphing Functions
Use the trapezoidal approximation to evaluate the following integral:
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:
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:
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:
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:
To evaluate the definite integral using the trapezoidal approximation, we must use the following formula:
Using the above formula, we get
Example Question #3 : How To Find Trapezoidal Approximation By Graphing Functions
Evaluate the integral using the trapezoidal approximation:
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:
To evaluate the definite integral using the trapezoidal approximation, the following formula is used:
Using the above formula, we get
.