Trapezoidal Approximation - CLEP Calculus

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Question

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

Answer

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:

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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Question

Is the following function increasing or decreasing at the point ?

Answer

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.

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Question

Use the trapezoidal approximation to approximate the following integral:

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

Answer

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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Question

Use the trapezoidal approximation to evaluate the following integral:

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

Answer

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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Question

Use the trapezoidal approximation to evaluate the following integral:

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

Answer

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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Question

Evaluate the following integral using the trapezoidal approximation:

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

Answer

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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Question

Evaluate the integral using the trapezoidal approximation:

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

Answer

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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Question

Evaluate the integral using the trapezoidal approximation:

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

Answer

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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Question

Evaluate the integral using the trapezoidal approximation:

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

Answer

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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Question

Evaluate the integral using the trapezoidal approximation:

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

Answer

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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Question

Evaluate the integral using the trapezoidal approximation:

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

Answer

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

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

Using the formula, we get

$(2)(\pi)(\frac{e^{-1}+e^1}{2})=\pi(e^{-1}+e^1)$

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Question

Integrate using the trapezoidal approximation:

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

Answer

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

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

Now, using the above formula, we can approximate the integral:

$\int_{-2}^{2}(-x^2+e^{2x})dx\approx (2+2)(\frac{-4+e^{4}-4+e^{-4}}{2})=-16+2e^4+2e^{-4}$

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Question

Evaluate the following integral using the trapezoidal approximation:

$\int_{a}^{b}5x^2dx$

where a and b are constants,

Answer

To solve the integral, it is easiest to first rewrite it so that the larger value is the upper bound. This is done by switching the bounds and also making the integral negative:

$-\int_{b}^{a}5x^2dx$

Now, we can use the trapezoidal approximation, which states that for a definite integral:

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

Using the above formula for our integral, we get

$-(a-b)(\frac{5a^2+5b^2}{2})$

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Question

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

Answer

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:

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

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

Compare your answer with the correct one above

Question

Is the following function increasing or decreasing at the point ?

Answer

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.

Compare your answer with the correct one above

Question

Use the trapezoidal approximation to approximate the following integral:

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

Answer

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

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

Compare your answer with the correct one above

Question

Use the trapezoidal approximation to evaluate the following integral:

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

Answer

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

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

Compare your answer with the correct one above

Question

Use the trapezoidal approximation to evaluate the following integral:

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

Answer

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

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

Compare your answer with the correct one above

Question

Evaluate the following integral using the trapezoidal approximation:

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

Answer

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:

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

Compare your answer with the correct one above

Question

Evaluate the integral using the trapezoidal approximation:

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

Answer

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

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

Compare your answer with the correct one above

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