LESSON 6.13 EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
1.
Find the derivatives of the functions defined by the following integrals
𝑥𝑥 sin 𝑡𝑡
(a) ∫0
1
𝑡𝑡
𝑥𝑥
𝑥𝑥 2 1
2
(d) ∫0 𝑒𝑒 tan 𝑡𝑡 𝑑𝑑𝑑𝑑
√𝑥𝑥 𝑠𝑠 2
(g) ∫1
2.
𝑠𝑠 2 +1
(e) ∫𝑥𝑥
2𝑡𝑡
cos 𝑥𝑥
(h) ∫−5
𝑑𝑑𝑑𝑑
cos 𝑥𝑥 1
2
(c) ∫1
𝑑𝑑𝑑𝑑
(f) ∫𝑥𝑥 cos 𝑡𝑡 2 𝑑𝑑𝑑𝑑
𝑡𝑡 cos 𝑡𝑡 3 𝑑𝑑𝑑𝑑
(i) ∫tan 𝑥𝑥 sin 𝑡𝑡 4 𝑑𝑑𝑑𝑑
(b) ∫0 𝑒𝑒 −𝑡𝑡 𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑡𝑡
𝑑𝑑𝑑𝑑
2
17
The graph of a function 𝑓𝑓 consists of a semicircle and two line-segments as shown. Let
𝑥𝑥
𝑔𝑔 be the function given by 𝑔𝑔(𝑥𝑥) = ∫0 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑.
(a) Find 𝑔𝑔(0), 𝑔𝑔(3), 𝑔𝑔(2), and 𝑔𝑔(5).
(b) Find all values of 𝑥𝑥 on the open interval (−2,5) at which 𝑔𝑔 has a relative
maximum. Justify your answers.
(c) Find the absolute minimum value of 𝑔𝑔 on the closed interval [−2, 5] and the value of 𝑥𝑥 at which it occurs. Justify your
answer.
(d) Write an equation for the line tangent to the graph of 𝑔𝑔 at 𝑥𝑥 = 3
(e) Find the 𝑥𝑥-ccoordinate of each point of inflection of the graph of 𝑔𝑔 on the open interval (−2,5). Justify your answer.
(f) Find the range of 𝑔𝑔.
3.
𝑥𝑥
Let 𝑔𝑔(𝑥𝑥) = ∫0 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑, where 𝑓𝑓 is the function whose graph is shown.
(a) Find 𝑔𝑔(0), 𝑔𝑔(1), 𝑔𝑔(2), and 𝑔𝑔(6).
(b) On what interval is 𝑔𝑔 increasing?
(c) Where does 𝑔𝑔 have a maximum value? What is the maximum value?
(d) Where does 𝑔𝑔 have a minimum value? What is the minimum value?
(e) Sketch a rough graph of 𝑔𝑔 on [0,7]
4.
𝑥𝑥
Let 𝑔𝑔(𝑥𝑥) = ∫−3 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑, where 𝑓𝑓 is the function whose graph is shown.
(a) Evaluate 𝑔𝑔(−3) and 𝑔𝑔(3).
(b) At what values of 𝑥𝑥 is 𝑔𝑔 increasing? Justify.
(c) At what values of 𝑥𝑥 does 𝑔𝑔 have a maximum value? Justify.
(d) At what values of 𝑥𝑥 does 𝑔𝑔 have a minimum value? Justify.
5.
Find the equation of the tangent line to the curve 𝑦𝑦 = 𝐹𝐹(𝑥𝑥) where
𝑥𝑥 3
𝐹𝐹(𝑥𝑥) = ∫1 √𝑡𝑡 2 + 7𝑑𝑑𝑑𝑑 at the point on the curve where 𝑥𝑥 = 1.
6.
Suppose that 5𝑥𝑥 3 + 40 = ∫𝑐𝑐 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑
(a) What is 𝐹𝐹(𝑥𝑥)?
𝑥𝑥
(b) Find the value of 𝑐𝑐
7.
𝑥𝑥
If 𝐹𝐹(𝑥𝑥) = ∫−1(𝑡𝑡 − 1)2 (𝑡𝑡 + 3) 𝑑𝑑𝑑𝑑 for what value of 𝑥𝑥 is 𝐹𝐹 decreasing? Justify your answer.
8.
𝑥𝑥
Let 𝐻𝐻(𝑥𝑥) = ∫0 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑 where 𝑓𝑓 is the continuous function with domain [0, 12] shown
to the right.
(a) Find 𝐻𝐻(0)
(b) On what interval of 𝑥𝑥 is 𝐻𝐻 increasing? Justify your answer.
(c) On what interval of 𝑥𝑥 is 𝐻𝐻 concave up? Justify your answer.
(d) Is 𝐻𝐻(12) positive or negative? Explain.
(e) For what value of 𝑥𝑥 does 𝐻𝐻 achieve its maximum value? Explain.
9.
𝑥𝑥
The graph of a function 𝑓𝑓 consists of a semicircle and two line-segments as shown. Let 𝑔𝑔(𝑥𝑥) = ∫1 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑.
(a) Find 𝑔𝑔(1), 𝑔𝑔(3), and 𝑔𝑔(−1).
(b) On what interval(s) of 𝑥𝑥 is 𝑔𝑔 decreasing? Justify your answer
(c) Find all values of 𝑥𝑥 on the open interval (−3,4) at which 𝑔𝑔 has a relative
maximum. Justify your answers.
(d) Find the absolute maximum value of 𝑔𝑔 on the interval (−3,4) and the value of 𝑥𝑥
at which it occurs. Justify your answer.
(e) On what interval(s) of 𝑥𝑥 is 𝑔𝑔 concave up? Justify your answer
(f) For what value(s) of 𝑥𝑥 does the graph of 𝑔𝑔 have an inflection point? Justify your
answer.
𝑥𝑥
10. The function 𝑔𝑔 is definited on the interval [0,6] by 𝑔𝑔(𝑥𝑥) = ∫1 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑 where 𝑓𝑓 is the function graphed in the figure.
(a) For what values of 𝑥𝑥 on the interval (0,6), does 𝑔𝑔 have a relative maximum?
Justify your answer.
(b) For what values of 𝑥𝑥 is the graph of 𝑔𝑔 concave down? Justify your answer.
(c) Write an equation for the tangent line to 𝑔𝑔 at the point where 𝑥𝑥 = 3.
(d) Sketch a graph of the function 𝑔𝑔. List the coordinates of all critical points and
inflection points.
𝑥𝑥 2
11. The function 𝐹𝐹 is defined for all 𝐹𝐹(𝑥𝑥) = ∫1 √𝑡𝑡 2 + 8 𝑑𝑑𝑑𝑑.
(a) Find 𝐹𝐹′(𝑥𝑥)
(b) Find 𝐹𝐹′(1)
(c) Find 𝐹𝐹′′(𝑥𝑥)
(d) Find 𝐹𝐹′′(1)
12. The graph of the velocity 𝑣𝑣(𝑡𝑡), in ft/sec, of a car traveling on a straight road
for 0 ≤ 𝑡𝑡 ≤ 35, is shown in the figure.
(a) Find the average acceleration of the car, in ft/sec2, over the interval 0 ≤ 𝑡𝑡 ≤ 35.
(b) Find an approximation for the acceleration of the car, in ft/sec2, at 𝑡𝑡 = 20. Show
your computations.
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(c) Approximate ∫5 𝑣𝑣(𝑡𝑡) 𝑑𝑑𝑑𝑑 with a Riemann sum, using the midpoints of three
subintervals of equal length. Explain the meaning of this integral.