Calculus 1 : Functions
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Example Questions
Example Question #2671 : Calculus
Determine the intervals on which the function is concave down:
To determine the intervals on which the function is concave down, we must determine the intervals on which the second derivative of the function is negative.
To start, we must find the second derivative:
The derivatives were found using the following rules:
Next, we must find the values at which the second derivative is equal to zero:
Using these values, we can make our intervals on which we check the sign of the second derivative:
Next, plug in any point on each interval into the second derivative function and check the sign. On the first interval, the second derivative is negative, on the second, it is positive, and on the third, it is negative. The intervals on which the second derivative is negative are concave down:
Example Question #1 : Trapezoidal Approximation
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 #202 : Graphing Functions
Is the following function increasing or decreasing at the point 
h(x) is increasing at 
h(x) is decreasing at 
h(x) is increasing at 
h(x) is decreasing at 
h(x) is decreasing at
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 #2681 : Calculus
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 #1 : 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 #205 : 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 #84 : Intervals
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 #2681 : Calculus
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
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