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MATH 1010U Lecture Notes - Lecture 35: Antiderivative

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silvermanatee152

23 Nov 2017

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1. let the more complicated one (and the one that when differentiated, leads to the qn*dx) be u: differentiate that u to get du/dx, substitute 2 into equation to find the equation in u, differentiate normally. 2 methods to integrate for definite integral: make the indefinite integral into a definite one. Substitute the boundaries at the final result. Integrate the equation as if it does not have boundaries (do not change the boundaries to u, leave it as it is) Do it as if = cosx(sinx + 1) 3. When you substitute the boundaries back, you can ignore the c" (it"s a constant that will cancel off: rewrite the boundary in u, integrate with boundaries intact. In this case, we let u = g(x) Substitute a and b into g(x) to find the new boundary (in u) Substitute such that the equation is __ du (__ in u")

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Calculus

ISBN: 9781319050733

Single Variable Calculus: Early Transcendentals

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ISBN: 9781337687805

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Related Questions

Trigonometric Substitution Rules

For v(a^2-u^2 ) let u = asin?? and v(a^2-u^2 ) = acos??
For v(a^2+ u^2 ) let u = atan? and v(a^2+u^2 ) = asec?
For v(u^2-a^2 ) let u = asec?? and v(u^2-a^2 ) = atan?

1. Integrate ?dx/v(25+4x^2 ). Identify the following:

a = u = du =

What adjustment, if any, needs to be made to the integral tocomplete the substitution in to u?

2. Rewrite the above integral so that it is all in u.

3. Which of the trig substitution rules above will we use to solvethis integral? Why? Draw and label the relevant right trianglediagram.

4. Use the rule you chose and the diagram to identify what you willreplace u, du and v(a^2+u^2 ) with.

5. Rewrite the integral from step 2 using the replacements fromstep 4. Did you use all the replacements? If not, why not? Can thisintegral be simplified? If so, do it?

6. Integrate the resulting trig function using a basicformula.

7. Use the information from steps 1 and 4 to rewrite the answer instep 6 in terms of the variable x.

u = 2x = 5tan? and v(a^2+u^2 )=v(25+4x^2 )=5sec?

8. How does knowing the trigonometric relationships as defined bythe three sides of a right triangle and the Pythagorean Theoremhelp us solve integrals using trig substitutions?

9. ?dx/(xv(49-9x^2 )) 10. ?dx/v(16x^2-1)

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help

Using the function y = f(x) obtained in (c), find the .y coordinate corresponding to the value x = -1 / 2 Evaluate both formulas for dy / dx at this point. Why do you think this happens.? Using the explicit derivative formula dy / dx = f (x) obtained in part (c) , find all points on the graph of y = f(x) where the tangent line is horizontal. Then substitute the x and y - coordinates of each such point into the implicit dy / dx formula given in part(b). Again, why does this happen? Again using the explicit derivative formula dy / dx = f(x) from part (c). find all points on the graph of f where the tangent slope equals 6. Now substitute the x- and y- coordinates of each such point into the implicit dy / dx formula given in part (b) to confirm this slope value. It is clear that no point (x, y) for which x + 2y = 0 will appear on the graph of original equation 2x = y - x2 / x +2y. In effect, points on the line x + 2y = 0 are "cut out" of the graph of the function f Relate this observation to the phenomena encountered earlier in parts (e) and (f). Once again referring to the explicit function f(x) obtained in part (c), where does the graph of f have a vertical asymptote? What happens when you substitute this X-value into the original implicit equation (given at the beginning of this worksheet) and attempt to solve for y ? Explain. [ Verification of the implicit derivative given in part (b).] Implicitly differentiate the equation 2x = y - x2 / x = 2y, applying the quotient role on the right-hand side. Clear the denominator and expand terms, then solve for dy / dx and simplify. Note: Solutions to this problem set are t0 be established by traditional algebraic or arithmetic means. using Maple software to verify each result To download the Maple template file prepared to assist you w ith these exercises, go to https;//my,ltu edu and login, select Calculus 1 and click on Workshop. Saw this file to your laptop hard drive and exit the web Hand-generated solutions are to be fully documented In the workshop report including all steps requiring algebraic or calculus computations - merely listing the answers produce with Maple is not sufficient. Printouts of your Maple verifications are required as part of your workshop report. Consider the equation 2x = y -x2 / x+ 2y. Verify that the points .A(1, -1) and B(-2, 4/3) lie on the graph of this equation. In part (j) of this worksheet you will be asked to verify- the derivative formula by implicitly differentiating the equation given above. Evaluate this derivative at points A and B. Solve for y in the original equation, expressing y as an explicit function of x. (For later reference, we will call this function f(x).) Compute dy / dx from this expression and simplify , then evaluate this explicit version of dy / dx at both A aud B. Considering the results of part (b), are these the values you expected? Why or why not? Evaluate each of the two formulas for dy / dx (both implicit and explicit) at the point C(-1, 0). Can you explain your results?

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Using the function y = f(x) obtained in (c), find the .y coordinate corresponding to the value x = -1 / 2 Evaluate both formulas for dy / dx at this point. Why do you think this happens.? Using the explicit derivative formula dy / dx = f (x) obtained in part (c) , find all points on the graph of y = f(x) where the tangent line is horizontal. Then substitute the x and y - coordinates of each such point into the implicit dy / dx formula given in part(b). Again, why does this happen? Again using the explicit derivative formula dy / dx = f(x) from part (c). find all points on the graph of f where the tangent slope equals 6. Now substitute the x- and y- coordinates of each such point into the implicit dy / dx formula given in part (b) to confirm this slope value. It is clear that no point (x, y) for which x + 2y = 0 will appear on the graph of original equation 2x = y - x2 / x +2y. In effect, points on the line x + 2y = 0 are "cut out" of the graph of the function f Relate this observation to the phenomena encountered earlier in parts (e) and (f). Once again referring to the explicit function f(x) obtained in part (c), where does the graph of f have a vertical asymptote? What happens when you substitute this X-value into the original implicit equation (given at the beginning of this worksheet) and attempt to solve for y ? Explain. [ Verification of the implicit derivative given in part (b).] Implicitly differentiate the equation 2x = y - x2 / x = 2y, applying the quotient role on the right-hand side. Clear the denominator and expand terms, then solve for dy / dx and simplify. Note: Solutions to this problem set are t0 be established by traditional algebraic or arithmetic means. using Maple software to verify each result To download the Maple template file prepared to assist you w ith these exercises, go to https;//my,ltu edu and login, select Calculus 1 and click on Workshop. Saw this file to your laptop hard drive and exit the web Hand-generated solutions are to be fully documented In the workshop report including all steps requiring algebraic or calculus computations - merely listing the answers produce with Maple is not sufficient. Printouts of your Maple verifications are required as part of your workshop report. Consider the equation 2x = y -x2 / x+ 2y. Verify that the points .A(1, -1) and B(-2, 4/3) lie on the graph of this equation. In part (j) of this worksheet you will be asked to verify- the derivative formula by implicitly differentiating the equation given above. Evaluate this derivative at points A and B. Solve for y in the original equation, expressing y as an explicit function of x. (For later reference, we will call this function f(x).) Compute dy / dx from this expression and simplify , then evaluate this explicit version of dy / dx at both A aud B. Considering the results of part (b), are these the values you expected? Why or why not? Evaluate each of the two formulas for dy / dx (both implicit and explicit) at the point C(-1, 0). Can you explain your results?