Science:Math Exam Resources/Courses/MATH101/April 2017/Question 06 (a)
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Question 06 (a) 

Find the slope of the tangent line to the curve at the point . Simplify your answer completely. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Try to use the derivative formula for integrals (Fundamental theorem of calculus), and recall that the slope of tangent line is just the derivative at this point. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. According to the fundamental theorem of calculus, if we have following expression where is continuous real valued function on some interval , then its derivative is just . Now if we have expression like what is the derivative of ? we think above equation as follows Then by chain rule and fundamental theorem of calculus, we know that .
Now if we have expression like we can write it as a combination Therefore, using above formula we have
This formula can be seen in any calculus book which is very important for calculating the derivatives of integrals. Now we apply it in this question, in this case
So We know that the slope of tangent line at point is just the derivative of at this point. Especially, when the derivative is . 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. The slope of the tangent line containing is . To find , consider the function defined by Then we can rewrite as . By the Fundamental Theorem of Calculus, So, by the Chain Rule, Hence, plugging , we get So, the slope of the tangent line is 