Example: Symbolic Calculus I

Symbolics > Working with Symbolics > Calculus > Example: Symbolic Calculus I

Solve symbolic problems in calculus by evaluating the expression for the problem using the symbolic evaluation operator and adding keywords to modify the problem, if necessary.

Derivatives and Integrals

1. Find the derivative of an expression that contains the atan function.

2. Assign the result of a symbolic differentiation to a function, which can be useful if you do not have values for all the variables.

3. Take the derivative of function sin with a complex argument.

4. Use the keyword rectangular to display the result in the standard form of a+bi.

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  <math resultRef="22">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <derivative />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">sin</id>
            <apply>
              <plus />
              <apply>
                <mult />
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
                <imag symbol="i">1</imag>
              </apply>
              <id xml:space="preserve">y</id>
            </apply>
          </apply>
        </lambda>
        <degree>
          <placeholder />
        </degree>
      </apply>
      <command>
        <id xml:space="preserve">rectangular</id>
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <div />
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <neg />
                    <apply>
                      <pow />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <apply>
                        <neg />
                        <parens>
                          <apply>
                            <mult />
                            <real>2</real>
                            <id xml:space="preserve" labels="VARIABLE">x</id>
                          </apply>
                        </parens>
                      </apply>
                    </apply>
                  </apply>
                  <real>1</real>
                </apply>
              </parens>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">sin</id>
                <id xml:space="preserve" labels="*">y</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <real>2</real>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <apply>
                  <neg />
                  <id xml:space="preserve" labels="VARIABLE">x</id>
                </apply>
              </apply>
            </apply>
          </apply>
          <apply>
            <mult />
            <apply>
              <div />
              <apply>
                <plus />
                <apply>
                  <mult />
                  <apply>
                    <id xml:space="preserve" labels="FUNCTION">cos</id>
                    <id xml:space="preserve" labels="*">y</id>
                  </apply>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="CONSTANT">e</id>
                    <apply>
                      <neg />
                      <parens>
                        <apply>
                          <mult />
                          <real>2</real>
                          <id xml:space="preserve" labels="VARIABLE">x</id>
                        </apply>
                      </parens>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <id xml:space="preserve" labels="FUNCTION">cos</id>
                  <id xml:space="preserve" labels="*">y</id>
                </apply>
              </apply>
              <apply>
                <mult />
                <real>2</real>
                <apply>
                  <pow />
                  <id xml:space="preserve" labels="CONSTANT">e</id>
                  <apply>
                    <neg />
                    <id xml:space="preserve" labels="VARIABLE">x</id>
                  </apply>
                </apply>
              </apply>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

5. Find the indefinite integral of a function.

6. Evaluate the integral by specifying the limits of integration.

7. Evaluate the integral numerically.

8. Use the keyword float to display results with a greater accuracy.

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  <math resultRef="26">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <mult />
            <apply>
              <pow />
              <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
            </apply>
            <apply>
              <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">sin</id>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <real>1</real>
        </upperBound>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">float</id>
          <real>30</real>
        </sequence>
      </command>
      <symResult>
        <real>0.909330673631478617034602154687</real>
      </symResult>
    </symEval>
  </math>
</region>

9. Evaluate expressions that contain parameters in the integrand or limits of integration.

10. Evaluate a parameterized integral that behaves as a finite under certain conditions.

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  <math resultRef="30">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">t1</id>
          </boundVars>
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">e</id>
            <apply>
              <mult />
              <apply>
                <neg />
                <id xml:space="preserve">c</id>
              </apply>
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">t1</id>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <id xml:space="preserve" labels="CONSTANT">∞</id>
        </upperBound>
      </apply>
      <command>
        <sequence>
          <id xml:space="preserve">assume</id>
          <apply>
            <greaterThan />
            <id xml:space="preserve">c</id>
            <real>0</real>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <div />
          <real>1</real>
          <id xml:space="preserve" labels="*">c</id>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

11. Show that PTC Mathcad understands the fundamental theorem of calculus that states the following relationships between derivatives and integrals:

Limits

1. Evaluate the limits of the following expression:

2. Compute the left-handed and right-handed limits of h(x).