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Calculus 1 : How to find trapezoidal approximation by graphing functions

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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.

Graph1

Possible Answers:

$A \approx 38.4$

$A\approx 29.6$

$A \approx 24.4$

$A\approx 19.2$

Correct answer:

$A\approx 29.6$

Explanation:

The trapezoid rule states that

$\\ A\approx \frac{\Delta x}{2}[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)]\\ \Delta x=\frac{b-a}{n}, x_i=a+i\Delta x$ .

Therefore, using our graph, we have:

$\\ \Delta x=\frac{8}{4}=2\\ \\ x_i=0,2,4,6,8$

We find the function values at the sample point:

f(x_i)=6,4,4,2.4,2.8

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

$A\approx\frac{2}{2}[6+2(4+4+2.4)+2.8]=29.6$

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Example Question #2 :Trapezoidal Approximation

Is the following function increasing or decreasing at the point?

h(x)=-x^5+5x^4+2x

Possible Answers:

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.

Correct answer:

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

Explanation:

Is the following function increasing or decreasing at the point?

h(x)=-x^5+5x^4+2x

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:

h(x)=-x^5+5x^4+2x

Becomes:

h'(x)=-5x^4+20x^3+2

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

h'(x)=-5(-6)^4+20(-6)^3+2=-43198

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

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Example Question #3 :Trapezoidal Approximation

Use the trapezoidal approximation to approximate the following integral:

$\int_{0}^{5}(x^2+2x+1)dx$

Possible Answers:

180

Correct answer:

Explanation:

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

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get

$5(\frac{36}{2})=90$

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Example Question #4 :Trapezoidal Approximation

Use the trapezoidal approximation to evaluate the following integral:

$\int_{1}^{3}(x^3+x^2+6x)dx$

Possible Answers:

Correct answer:

Explanation:

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

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get

$2(\frac{27+9+10-1-1-6}{2})=46$

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Example Question #5 :Trapezoidal Approximation

Use the trapezoidal approximation to evaluate the following integral:

$\int_{0}^{\pi} 3\cos(x)dx$

Possible Answers:

$3\pi$

$\frac{3\pi}{2}$

$-3\pi$

$6\pi$

Correct answer:

$-3\pi$

Explanation:

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

$\int_{a}^{b}f(x)dx= (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get

$\pi(\frac{-3-3}{2})=-3\pi$

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Example Question #6 :Trapezoidal Approximation

Evaluate the following integral using the trapezoidal approximation:

$\int_{0}^{\2\pi}\frac{e^{2x}}{5\cos(x)}dx$

Possible Answers:

$2\pi (\frac{e^{4\pi}-1}{5})$

$2\pi (\frac{e^{4\pi}-1}{10})$

$\pi (\frac{e^{4\pi}-1}{5})$

$2\pi (e^{4\pi}-1)$

$2\pi (\frac{e^{4\pi}+1}{5})$

Correct answer:

$2\pi (\frac{e^{4\pi}-1}{10})$

Explanation:

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

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)-f(a)}{2})$

Using the above formula, we get the following:

$2\pi(\frac{e^{4\pi}-1}{10})$

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Example Question #7 :Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{5}^{10}( x^2+e^{x-5})dx$

Possible Answers:

$\frac{5}{2}(75+e^5)$

5(126+e^5)

$\frac{5}{2}(126-e^5)$

$\frac{5}{2}(126+e^5)$

Correct answer:

$\frac{5}{2}(126+e^5)$

Explanation:

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

$\int_{a}^{b} f(x)dx\approx \frac{(b-a)(f(a)+f(b))}{2}$

So, using the above formula, we get

$\frac{(5)(100+e^5+26)}{2}$

which simplifies to

$\frac{5}{2}(126+e^5)$

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Example Question #8 :Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{2}^{4}\frac{ t^3-t^2}{t}dt$

Possible Answers:

Correct answer:

Explanation:

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

$\int_{a}^{b}f(x)dx\approx (b-a)\frac{(f(a)+f(b))}{2}$

Using the above formula, we get

$\frac{2(\frac{48}{4}+\frac{4}{2})}{2}=14$

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Example Question #9 :Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{-2}^{2}\frac{x^2+e^{x-2}}{3x}dx$

Possible Answers:

$\frac{9+e^{-4}}{3}$

$\frac{1+e^{-4}}{3}$

$\frac{1-e^{-4}}{3}$

$\frac{1-e^{-4}}{6}$

Correct answer:

$\frac{1-e^{-4}}{3}$

Explanation:

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

$\int_{a}^{b} f(x)dx\approx (b-a)\frac{(f(b)+f(a))}{2}$

Using the formula, we get

$\frac{4(\frac{5}{6}+\frac{4+e^{-4}}{-6})}{2}=\frac{1-e^{-4}}{3}$

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Example Question #10 :Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{0}^{10} (\frac{x^2e^{10-x}}{5}+5x)dx$

Possible Answers:

150

350

250

700

Correct answer:

350

Explanation:

To evaluate the definite integral using the trapezoidal approximation, the following formula is used:

$\int_{a}^{b}f(x)dx\approx (b-a)\frac{(f(b)+f(a))}{2}$

Using the above formula, we get

$\frac{10[(\frac{100}{5}+50)+(0)]}{2}=350$ .

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Calculus 1 : How to find trapezoidal approximation by graphing functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

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

Example Question #2 :Trapezoidal Approximation

Example Question #3 :Trapezoidal Approximation

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