Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyFTC-ITheorem: Fundamental Theorem of Calculus - Part IActivity 11HomeworkCC BY-NC-SA 4.0

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 4.9**– Antiderivatives**Section 5.3**– The Fundamental Theorem of Calculus**Section 5.4**– Indefinite Integrals and the Net Change Theorem

**Objective 01–01**: I can state the*Fundamental Theorem of Calculus*.**Objective 01–02**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–03**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–04**: I can determine the properties of an area function using*FTC-I*and*Calculus I*.**Objective 01–05**: I can evaluate indefinite integrals using the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–06**: I can evaluate definite integrals using the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–07**: I can use the*Net Change Theorem*to identify the meaning of .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **Objective 01–08**: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.**Objective 01–09**: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Recall, from *CPT_01c_FTC_I.html*,

**Fundamental Theorem of Calculus - Part I**: Let

**Note**: The variable **independent** variable, and the variable *dummy* variable.

**Conditions to be met**:

is continuous for all values on$f(t)$ ;$[a,b]$ is defined as the integral$F(x)$ , where${\int}_{a}^{x}f(t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ a constant,

, is the lower limit;$a$ the variable,

, and not some function of$x$ , is the upper limit;$x$ the integral is in terms of a dummy variable, e.g.,

, and not$t$ .$x$

**Conclusion of FTC-I**:

If the above conditions are met, then the conclusion of the FTC-I is

Use what you learned from the previous **Activity** page (*ACT_01e_Derivative_of_Integral_Function.html*). Without using any technology, differentiate the following:

. Explain.$g(x)={\int}_{2}^{x}\sqrt{{t}^{2}+t-1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ *Check your answer with the embedded quiz below*. . Explain.$h(x)={\int}_{x}^{1}\frac{t+1}{{t}^{2}+1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ *Check your answer with the embedded quiz below*. . Explain.$k(x)={\int}_{1}^{{x}^{2}}\mathrm{tan}(t+1){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ *Check your answer with the embedded quiz below*. . Explain.$m(x)={\int}_{1/x}^{-1}\mathrm{cos}(2-t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$

At this time, you should be able to complete the following assignments:

**Section 5.3**: # 7, 9, 11, 13, 15, 17, 19.

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**Created**: Tuesday, 25 August 2020 03:49 EDT
**Last Modified**: Monday, 30 May 2022 - 13:48 (EDT)