Barrons AP Calculus - David Bock

where r is the radius and h the height. We seek to minimize S, the total surface area, where Solving (1) for h, we have which we ...

lay the pipe in the river but only $2 per foot to lay it on land. (a) Express the cost of laying the pipe in terms of u. (b) How ...

So u = yields minimum cost. Thus, the pipe can be laid most economically if some of it is laid in the river from the source S to ...

EXAMPLE 25A Given the graph of f (x) shown in Figure N4–12, sketch f ′(x). FIGURE N4–12 Point x = Behavior of f Behavior of f ′ ...

Given the graph of f ′(x) shown in Figure N4–13, sketch a possible graph of f. FIGURE N4–13 SOLUTION: First, we note that f ′(−3 ...

If a particle moves along a line according to the law s = f (t), where s represents the position of the particle P on the line a ...

(d) The speed when equals FIGURE N4–15 (e) P’s motion can be indicated as shown in Figure N4–15. P moves to the right if t < ...

t: 0 3 4 s: 0 −27 0 The particle travels a total of 54 units between t = 0 and t = 4. (Compare with Example 13, where the functi ...

and its magnitude is the vector’s length: where we have used ax and ay for respectively. Examples 28 and 29 are BC ONLY. EXAMPLE ...

FIGURE N4–18 (e) For the speed |v| at any time t We see immediately that the speed is a maximum when and a minimum when t = 0 or ...

FIGURE N4–19a FIGURE N4–19b L. TANGENT-LINE APPROXIMATIONS Local linear approximation If f ′(a) exists, then the local linear ap ...

FIGURE N4–20 where f (a) + f ′(a)(x − a) is the linear or tangent-line approximation for f (x), and f ′(a)(x − a) is the approxi ...

EXAMPLE 32 A very useful and important local linearization enables us to approximate (1 + x) k by 1 + kx for k any real number a ...

EXAMPLE 35 Suppose the diameter of a cylinder is 8 centimeters. If its circumference is increased by 2 centimeters, how much lar ...

At the instant in question, u = 6, v = 8, and z = 10, so EXAMPLE 37 The diameter and height of a paper cup in the shape of a con ...

FIGURE N4–23 SOLUTION: Liquid flowing in at a constant rate means the change in volume is constant per unit of time. Obviously, ...

FIGURE N4–24 BC ONLY SOLUTIONS: (a) Use r = 2(1 + cos θ), x = r cos θ, y = r sin θ, and r ′ = −2 sin θ; then At (b) Since the ca ...

For BC Calculus students, this chapter reviewed finding slopes of curves defined parametrically or in polar form. We have also r ...

The minimum value of the slope of the curve y = x^5 + x^3 − 2x is (A) 0 (B) 2 (C) 6 (D) −2 (E) none of these The equation of th ...

If the side e of a square is increased by 1%, then the area is increased approximately (A) 0.02e (B) 0.02e^2 (C) 0.01e^2 (D) 1% ...