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Calculus 1 : Intervals

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Example Questions

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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:

350

700

250

150

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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Example Question #11 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

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

Possible Answers:

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

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

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

Correct answer:

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

Explanation:

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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Example Question #12 : Trapezoidal Approximation

Integrate using the trapezoidal approximation:

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

Possible Answers:

$-16+2e^4+2e^{-4}$

$-8+e^4+e^{-4}$

$2e^4-2e^{-4}$

Correct answer:

$-16+2e^4+2e^{-4}$

Explanation:

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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Example Question #93 : Intervals

Evaluate the following integral using the trapezoidal approximation:

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

where a and b are constants, b<a

Possible Answers:

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

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

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

(b-a)(5a^2+5b^2)

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

Correct answer:

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

Explanation:

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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Example Question #1 : Continuous On The Interval

On which of the following intervals is the function $$ f(x)=\frac{x^2-4}{x+2}$ continuous?

Possible Answers:

$(-\infty,-2)$

$(-\infty,0)$

(-10,10)

$(-3,\infty)$

(-6,6)

Correct answer:

$(-\infty,-2)$

Explanation:

The function has a removable discontinuity at x=-2 .

Since this function is undefined at x=-2 is it not continuous across any interval containing x=-2 .

Notice that the correct answer is an open interval that goes up to, but does not include .

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Example Question #1 : How To Find Continuous On The Interval By Graphing Functions

Describe the function

$f(x)=\left\{\begin{matrix}-\sqrt{x}\textup{ when }x\geq 0 \\ -\sqrt{-x}\textup{ when }x\leq 0 \end{matrix}\right.$

on the interval [-4, 4] .

Possible Answers:

Continuous; Non-differentiable

Non-continuous; Differentiable

Non-continuous; Non-differentiable

Continuous; Differentiable

Correct answer:

Continuous; Non-differentiable

Explanation:

This function (shown below) is defined for every value along the interval with the given conditions (in fact, it is defined for all real numbers), and is therefore continuous. However, there is a cusp point at (0, 0), and the function is therefore non-differentiable at that point.

Function

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Example Question #1 : How To Find Continuous On The Interval By Graphing Functions

On what interval is the derivative of the function:

$f(x)=\sqrt{x-3}$

continuous?

Possible Answers:

$[3, \infty)$

$(-\infty, 3]$

The function is not continuous.

$(-\infty, 3)$

$(3, \infty)$

Correct answer:

$(3, \infty)$

Explanation:

The derivative of the funtion using the power rule

$\frac{d}{dx}x^n=nx^{n-1}$ is $\frac{1}{2(\sqrt{x-3})}$ ,

so it is not continuous when x-3=0 or is negative.

This occurs when $x\leq3$ , so the interval of continuity will be when x>3 , which is the interval $(3, \infty)$ .

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Calculus 1 : Intervals

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #7 : Trapezoidal Approximation

Example Question #8 : Trapezoidal Approximation

Example Question #9 : Trapezoidal Approximation

Example Question #10 : Trapezoidal Approximation

Example Question #11 : Trapezoidal Approximation

Example Question #12 : Trapezoidal Approximation

Example Question #93 : Intervals

Example Question #1 : Continuous On The Interval

Example Question #1 : How To Find Continuous On The Interval By Graphing Functions

Example Question #1 : How To Find Continuous On The Interval By Graphing Functions

All Calculus 1 Resources

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