Barrons AP Calculus - David Bock

When using parametric equations we must be sure to express everything in terms of the parameter. In Example 22 we replaced in te ...

FIGURE N6–1 *It is not necessary that the subintervals be of equal length, but the formulation is generally simpler if they are. ...

FIGURE N6–3 Note that if f is continuous then the area between the graph of f on [a, b] and the x-axis is given by This implies ...

(b) the right sum, (c) the midpoint sum, and (d) the integral. SOLUTIONS: Here (a) For a left sum we use the left-hand altitudes ...

sums are also sometimes called “rules.”) E2. Using Trapezoids. We now find the areas of the strips in Figure N6–6 by using trape ...

SOLUTION: Note that the subinterval widths are not equal. (a) In each subinterval, we sketch the rectangle with height determine ...

If f is an increasing function on [a,b], then while if f is decreasing, then From Figure N6–7 we infer that the area of a trapez ...

FIGURE N6–9 † Note that the trapezoid AT 1 T 2 B is different from the trapezoids in Figure N6–7, which are like the ones we use ...

1, the sign of f ′ changes from positive to negative; as x passes through 3, the sign of f ′ changes from negative to positive. ...

So the function f (x) has a local maximum at (1,2), a point of inflection at (2,1.2), and a local minimum at (3,0.4) where we ha ...

It is left to the student to complete the missing steps above and to verify the distances in the following table (t = time in mi ...

Here is the completed table: x 0123456 F(x) 013431 −0.5 EXAMPLE 31 The graph of the function f(t) is shown in Figure N6–14. FIGU ...

EXAMPLE 32 Graphs of functions f (x), g(x), and h(x) are given in Figures N6–15a, N6–15b, and N6–15c. Consider the following sta ...

FIGURE N6–15c EXAMPLE 33 Assume the world use of copper has been increasing at a rate given by f (t) = 1.5e0.015t, where t is me ...

F. INTERPRETING ln x AS AN AREA It is quite common to define ln x, the natural logarithm of x, as a definite integral, as follow ...

under above the t-axis, and bouned at the left by t = 1 and at the right by t = 2 (the shaded region in Figure N6–16). Since is ...

FIGURE N6–17 EXAMPLE 37 Find the average value of f (x) = ln x on the interval [1,4]. SOLUTION: EXAMPLE 38 Find the average valu ...

FIGURE N6–18 SOLUTIONS: (1) (d), since acceleration is the derivative of velocity and in (d) v ′, the slope, is constant. (2) (e ...

FIGURE N6–19b SOLUTIONS: See Figure N6–19b. (a) f (b) − f (a) = length RQ. (b) = slope of secant PQ. (c) F(b) − F(a) = = area of ...

maximum speeds; it equals or 55 mi for the total distance. (b) The acceleration, which is the slope of v(t), appears greatest at ...