Fundamental - Antiderivatives and Areas Name Math 125 Quiz Section In this worksheet we explore the Fundamental Theorem of Calculus and applications of

Fundamental - Antiderivatives and Areas Name Math 125 Quiz...

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Unformatted text preview: Antiderivatives and Areas Name Math 125 Quiz Section In this worksheet, we explore the Fundamental Theorem of Calculus and applications of the Area Problem to problems involving distance and velocity. We also consider integrals involving net and total change. FTC Practice 1 Let f (x) be given by the graph to the right and define x 3 A(x) = 0 f (t) dt. Compute the following. 2 y=f(x) A(1) = A(3) = A (1) = A (3) = A(2) = 1 A(4) = A (2) = A (4) = 1 2 3 4 5 The maximum value of A(x) on the interval [0, 5] is The maximum value of A (x) on the interval [0, 5] is Velocity and Distance 2 A toy car is travelling on a straight track. Its velocity v(t), in m/sec, be given by the graph to the right. Define s(t) to be the position of the car in meters. Choose coordinates so that s(0) = 0. Compute the following. s(2) = v(2) = s(4) = v(4) = s(6) = 1 2 3 4 5 6 7 3 2 y=v(t) 1 v(6) = -1 The maximum value of s(t) on the interval [0, 7] is The minimum value of s(t) on the interval [0, 7] is The maximum value of v(t) on the interval [0, 7] is The minimum value of v(t) on the interval [0, 7] is Net and Total Change 2 2 3 (a) Evaluate −2 x2 − 4 dx and −2 x2 − 4 dx and explain your answers. 3 3 (b) Now evaluate −3 x2 − 4 dx and −3 x2 − 4 dx and explain your answers. ¦ ¥£ ¢ £¢ ¢ ¢ ¢ ¢ ¢£¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦£ ¤¤£¤£¤£¤£¤£¤£¤£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ ¥£ ¢£¤£¤£¤£¤£¤£¤£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ £¢£¢£¢£¢£¢£¢£¤ ¡¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ ¦£ £¤£¤£¤£¤£¤£¤£¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¢¤£¢£¢£¢£¢£¢£¢£¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ ¦¥£ £¤£¤£¤£¤£¤£¤£¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¥£ ¢¤£¢£¢£¢£¢£¢£¢£¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ ¦£ £¤£¤£¤£¤£¤£¤£¢¤¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¥¦ ¥£ ¦£ ¥£ ¢¢¤£¢£¢£¢£¢£¢£¢£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ £¢£¢£¢£¢£¢£¢£¤¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¥¦ ¦£ ¤¤£¤£¤£¤£¤£¤£¤£¢¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ ¥£ ¢£¤£¤£¤£¤£¤£¤£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ £¢£¢£¢£¢£¢£¢£¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ ¦£ £¤£¤£¤£¤£¤£¤£¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¥£ ¢¤£¢£¢£¢£¢£¢£¢£¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ ¦£ £¤£¤£¤£¤£¤£¤£¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¥£ ¢¤£¢£¢£¢£¢£¢£¢£¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ ¦£ £¤£¤£¤£¤£¤£¤£¢¤¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¥¦ ¥£ ¦£ ¥£ ¢¢¤£¢£¢£¢£¢£¢£¢£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ £¢£¢£¢£¢£¢£¢£¤££££££££££££££££¥¦ ¦£ ¤¤£¤£¤£¤£¤£¤£¤£¢¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¥£ ¢£¤£¤£¤£¤£¤£¤£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ £¢£¢£¢£¢£¢£¢£¤¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¥¦ £¤£¤£¤£¤£¤£¤£¢¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ ¢¤£¢£¢£¢£¢£¢£¢£¢¤¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦ ¦¥¦£ ¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦ ¢¤£¢¤£¢¤£¢¤£¢¤£¢¤¤¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦ ¦¦¥¦¥ ¤£££££££¢¤¢¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦ ¢¢¤£¤¤¢£¤¤¢£¤¤¢£¤¤¢£¤¤¢£¤¤¢£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¢¤¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£¥¦£ £¤£¤£¤£¤£¤£¤£¤¢££££££££££££££££¥¦ ¦¥£ ¥£ ¢¢¤£¤£¤£¤£¤£¤£¤£¢¤¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¥¦¦ ¦¥£ £¢£¢£¢£¢£¢£¢£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£ ¥£ £¢£¢£¢£¢£¢£¢£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¦£ £¢£¢£¢£¢£¢£¢£¢¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£¦£ ¥£ £ ¢¤£¤£¤£¤£¤£¤£¤£¤¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥£¥¦ £¢£¢£¢£¢£¢£¢£¤¢££££££££££££££££¥¦ ¤¢¤£¤£¤£¤£¤£¤£¤£¢ ¢£¢£¢£¢£¢£¢£¢£ £¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¤¢ ¤¢¤£¤£¤£¤£¤£¤£¤£¢ ¢£¢£¢£¢£¢£¢£¢£ £¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¤¢ ¤¢¤£¤£¤£¤£¤£¤£¤£¢ ¢£¢£¢£¢£¢£¢£¢£ £¤£¤£¤£¤£¤£¤£¤ £¢£¢£¢£¢£¢£¢£¢ ¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤ ¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¤¢¤ ¤¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤¢ ¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤£¢¤ ¤¢£¤£¤£¤£¤£¤£¤£ ¢£¢£¢£¢£¢£¢£¢£ £££££££¢¢¤¤ 1 2 4 An artist you know wants to make a figure consisting of the region between the curve y = x2 and the x-axis for 0≤x≤3 Another Area Problem (i) Where should the artist divide the region with a vertical line so that each piece has the same area? (See the picture.) (ii) Where should the artist divide the region with vertical lines to get 3 pieces with equal areas? 10 0 2 4 6 8 y=x2 ? 3 ...
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