AP Calculus BC Study Guide – Flashcards

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Suppose that f''(x) exists on the interval (a,b), i) If f''(x) > 0 in (a,b), then f is concave (upward/downward) in (a,b). ii) If f''(x) < 0 in (a,b), then f is concave (upward/downward) in (a,b).

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i) upward ii) downward

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To locate the points of inflection of y = f(x), find...

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the points where f''(x) = 0 OR where f''(x) fails to exist. Then, test these points to make sure that f''(x) 0 on the other.

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If a function is differentiable at the point x = a, it (is/is not) continuous at that point. The converse is (true/false). Continuity (does/does not) imply differentiability.

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i) is ii) false iii) does not

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Euler's Method:

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Starting with the given point (x?,y?), the point (x? + ?x, y? + f'(x?,y?)?x) approximates a nearby point. (This approximation may then be used as the starting point to calculate a third point and so on.)

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Logarithm functions grow (slower/faster) than any power function (x?).

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slower

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Among power functions, those with higher powers grow (faster/slower) than those with lower powers.

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faster

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All power functions grow (faster/slower) than any exponential function (a^x, a > 1).

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slower

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Among exponential functions, those with larger bases grow (faster/slower) than those with smaller bases.

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faster

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We say, that as x??: f(x) grows (faster than/slower than/at the same rate as) g(x) if lim(x??) f(x) / g(x) = ? or if lim(x??) g(x) / f(x) = 0

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faster than

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We say, that as x??: f(x) grows (faster than/slower than/at the same rate as) g(x) if lim(x??) f(x) / g(x) = L ? 0.

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at the same rate as

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L'Hôpital's Rule:

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If lim(x?a) f(x) / g(x) is of the form 0 / 0 or ? / ?, and if lim(x?a) f'(x) / g'(x) exists, then lim(x?a) f(x) / g(x) = lim(x?a) f'(x) / g'(x).

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Two functions f and g are inverses of each other if...

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f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f.

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To test if a graph has an inverse, use the...

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horizontal line test

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If f is strictly increasing or decreasing in an interval, then f (has/does not have) an inverse.

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has

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i) If f is differentiable at every point on an interval I, and f'(x) ? 0 on I, then g = f?¹(x) (is/is not) differentiable at every point of the interior of the interval f(I). ii) If the point (a,b) is on f(x), then the point (b,a) (is/is not) on g = f?¹(x).

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i) is ii) is (Furthermore, g'(b) = 1 / f'(a).)

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Properties of y = e^x:

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i) y = e^x is the inverse function of y = ln x. ii) The domain is -? ; x 0. iv) d/dx e^x = e^x and d/dx e^f(x) = f'(x)*e^f(x). v) e^x? * e^x? = e^(x? + x?) vi) y = e^x is continuous, increasing, and concave up for all x. vii) lim(x??) e^x = ? and lim(x?-?) e^x = 0. viii) e^ln x = x, for x > 0 and ln(e^x) = x for all x.

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Properties of y = ln x:

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i) The domain is x > 0. ii) The range is -? < y <?. iii) y = ln x is continuous and increasing everywhere on its domain. iv) ln (a * b) = ln a + ln b. v) ln (a / b) = ln a - ln b. vi) ln a^r = r * ln a. vii) y = ln x < 0 if 0 < x < 1. viii) lim (x??) ln x = ? and lim(x?0?) ln x = -?. ix) log_a x = ln x / ln a x) d/dx ln f(x) = f'(x) / f(x) and d/dx ln x = 1 / x

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Trapezoidal Rule:

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If a function f is continuous on the closed interval [a,b] where [a,b] has been EQUALLY partitioned into n subintervals, then ?a?b f(x) dx ? ((b - a) / n))[f(x?) + 2f(x?) + 2f(x?) + ... + 2f(x_n-?) + f(x_n)], which is equivalent to (1 / 2)(Leftsum + Rightsum).

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Definition of a Definite Integral as the Limit of a Sum:

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Suppose that a function f(x) is continuous on the closed interval [a,b]. Divide the interval into n equal subintervals of length ?x = (b - a) / n. Choose one number in each subinterval, in other words, x? in the first, x? in the second, ..., k_k in the kth, ..., and x_n in the nth. Then, lim(n??) ?(k=1, n) f(x_k)?x = ?a?b f(x) dx = F(b) - F(a).

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Properties of the Definite Integral:

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Let f(x) and g(x) be continuous on [a,b]. i) ?a?b c * f(x) dx = c?a?b f(x) dx for any constant c. ii) ?a?a f(x) dx = 0. iii) ?a?b f(x) dx = -?b?a f(x) dx. iv) ?a?b f(x) dx = ?a?c f(x) dx + ?c?b f(x) dx, where f is continuous on an interval containing the numbers a, b, and c. v) If f(x) is an odd function, then ?-a?a f(x) dx = 0. vi) If f(x) is an even function, then ?-a?a f(x) dx = 2?0?a f(x) dx. vii) If f(x) ? 0 on [a,b], then ?a?b f(x) dx ? 0. viii) If g(x) ? f(x) on [a,b], then ?a?b g(x) dx ? ?a?b f(x) dx.

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Fundamental Theorem of Calculus:

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?a?b f(x) dx = F(b) - F(a), where F'(x) = f(x), or d/dx ?a?b f(x) dx = f(x).

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Second Fundamental Theorem of Calculus:

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d/dx ?a?x f(t) dt = f(x) or d/dx ?h(x)?g(x) f(t) dt = g'(x) * f(g(x) - h'(x) * f(h(x)).

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Explain "Velocity":

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The velocity of an object tells how fast it is moving and in what direction. Velocity is an instantaneous rate of change. If velocity is positive, then the object is moving away from its point of origin. If velocity is negative, then the object is moving back towards its point of origin. If velocity is 0, then the object is not moving at that time. The velocity of a function is found by taking its first derivative.

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Explain "Speed":

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The speed of an object is the absolute value of its velocity, |v(t)|, and tells how fast it is moving, disregarding its direction. The speed of a particle increases when the velocity and acceleration have the same signs. The speed decreases when the velocity and acceleration have opposite signs.

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Explain "Acceleration":

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The acceleration of an object is the instantaneous rate of change of velocity, that is, a(t) = v'(t). Negative acceleration means that the velocity is decreasing, and vice-versa for increasing acceleration. The acceleration of a function is found by taking its second derivative.

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Mean Value Theorem for Integrals:

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The average value of f(x) on [a,b] is (1 / (b - a)) * ?a?b f(x) dx.

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Area Between Curves:

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If f and g are continuous functions such that f(x) ? g(x) on [a,b], then the area between the curves is ?a?b f(x) - g(x) dx or ?a?b [Top - Bottom] dx or ?c?d [Right - Left] dx.

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Integration by Parts

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If u = f(x) and v = g(x), and if f'(x) and g'(x) are continuous, then ?u dx = uv - ?v du. (Note. the goal of the procedure is to choose u and dv such that ?v du is easier to solve than the original problem.)

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Volume of Solids of Revolution - Washer Method:

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Revolving around a horizontal line: V = ??a?b R?² - R?² dx Revolving around a vertical line (or use the Shell Method): V = ??c?d R?² - R?² dy (Remember to account for the distance between each radius and the axis of revolution if it is not x = 0 or y = 0.)

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Volume of Solids of Revolution - Shell Method:

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Revolving around a vertical line: V = 2??a?b r(x) * h(x) dx r(x) = x, if axis of revolution (a.r) is x = 0. r(x) = (x - a.r.), if a.r. is to the left of the region. r(x) = (a.r. - x), if a.r. is to the right of the region. h(x) = f(x), if only revolving with one function. h(x) = (top - bottom), if revolving the region between two functions.

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Logistic Growth:

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y = C / (1 + Ae^(-kt)). C = carrying capacity A = (C - initial) / initial k = constant (Note: k must be calculated using known values.)

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r² =

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x² + y²

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Definition of Arc Length:

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If the function given by y = f(x) represents a smooth curve on the interval [a,b], then the arc length of f between a and b is given by s = ?a?b ?(1 + [f'(x)]²) dx.

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?a?b f(x) dx is an improper integral if...

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i) f becomes infinite at one or more points of the interval of integration, or ii) one or both of the limits of integration is infinite, or iii) both (i) and (ii) hold.

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Parametric Form of the Derivative:

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First derivative: dy/dx = dy/dt / dx/dt, dt ? 0. Second derivative: d²y/dx² = d/dx [dy/dx] = d/dt [dy/dx] / dx/dt.

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Arc Length in Parametric Form:

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s = ?a?b ?([dx/dt]² + [dy/dt]²) dt = ?a?b ?([f'(t)]² + [g'(t)]²) dt

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Speed in Parametric Form:

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speed = ?([f'(t)]² + [g'(t)]²)

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Position Vector:

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r(t) =

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Velocity Vector:

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v(t) =

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Speed Vector:

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speed = |v(t)| = ?([dx/dt]² + [dy/dt]²)

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Acceleration Vector:

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a(t) =

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x = r(sin/cos) ? y = r(sin/cos) ?

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i) cos ii) sin

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dy/dx r(?) =

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(r'(?) * sin ? + r(?) * cos ?) / (r'(?) * cos ? - r(?) * sin ?)

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Length of a Curve:

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L = ?a?b ?[r² + (dr/d?)²] d?

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° =

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ac + bd

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Unit Vector:

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, |v| = magnitude = ?(a² + b²)

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Angle Between 2 Vectors:

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? = Arccos ([u ° v] / [|u| * |v|])

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Area Between a Curve and the Pole:

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A = (1 / 2) * ?a?b r² d?

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Area Between 2 Curves:

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A = (1 / 2) * ?a?b R² - r² d?

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Surface Area of Revolution:

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Revolving around the x-axis (? = 0): SA = ?a?b 2? * rsin ? * ?(r² + [dr/d?]²) d? Revolving around the y-axis (? = ? / 2): SA = ?a?b 2? * rcos ? * ?(r² + [dr/d?]²) d?

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If a line = , m =

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b / a.

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Vector Velocity:

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Velocity = (speed) * (direction) = |v| * (v / |v|).