AB Calculus Exam Review Solutions

Flashcards for AB Calculus Exam Review

Find the zeros of f(x).

Set function equal to 0. Factor or use quadratic equation if quadratic. Graph to find zeros on calculator.

Find the intersection of f(x) and g(x).

Set the two functions equal to each other. Find intersection on calculator.

Show that f(x) is even.

Show that f(x) = f(-x). This shows that the graph of f is symmetric to the y-axis.

Show that f(x) is odd.

Show that f(-x) = -f(x). This shows that the graph of f is symmetric to the origin.

Find the domain of f(x).

Assume domain is (-∞, ∞). Restrict domains: denominators ≠ 0, square roots of only non-negative numbers, logarithm or natural log of only positive numbers.

Find vertical asymptotes of f(x).

Express f(x) as a fraction, express numerator and denominator in factored form, and do any cancellations. Set denominator equal to 0.

If continuous function f(x) has f(a) < k and f(b) > k, explain why there must be a value c such that a < c < b and f(c) = k.

This is the Intermediate Value Theorem.

Find lim (x→a) f(x).

Step 1: Find f(a). If you get a zero in the denominator, Step 2: Factor numerator and denominator of f(x). Do any cancellations and go back to Step 1. If you still get a zero in the denominator, the answer is either ∞, -∞, or does not exist. Check the signs of lim (x→a-) f(x) and lim (x→a+) f(x) for equality.

Find lim (x→a) f(x) where f(x) is a piecewise function.

Determine if lim (x→a-) f(x) = lim (x→a+) f(x) by plugging in a to f(x), x < a and f(x), x > a for equality. If they are not equal, the limit doesn't exist.

Show that f(x) is continuous.

Show that 1) lim (x→a) f(x) exists, 2) f(a) exists, and 3) lim (x→a) f(x) = f(a)

Find lim (x→∞) f(x) and lim (x→-∞) f(x).

Express f(x) as a fraction. Determine location of the highest power: Denominator: lim (x→∞) f(x) = lim (x→-∞) f(x) = 0. Both Num and Denom: ratio of the highest power coefficients. Numerator: lim (x→∞) f(x) = +/- ∞ (plug in large number).

Find horizontal asymptotes of f(x).

lim (x→∞) f(x) and lim (x→-∞) f(x)

Find lim (h→0) using the derivative definition.

f(x + h) − f(x) / h or lim (x→a) f(x) − f(a) / x-a

Find the average rate of change of f on [a, b].

f(b)-f(a) / b-a

Find the instantaneous rate of change of f at x = a.

Find f'(a)

Given a chart of x and f(x) and selected values of x between a and b, approximate f'(c) where c is a value between a and b.

Straddle c, using a value of k ≥ c and a value of h ≤ c. f'(c) = f(k)-f(h) / k-h

Find the equation of the tangent line to f at (x₁, y₁).

Find slope m = f'(x). Then use point slope equation: y - y₁ = m ( x − x₁)

Find the equation of the normal line to f at (x₁, y₁).

Find slope m⊥ = -1 / f'(x). Then use point slope equation: y - y₁ = m ( x − x₁)

Find x-values of horizontal tangents to f.

Write f'(x) as a fraction. Set numerator of f'(x) = 0.

Find x-values of vertical tangents to f.

Write f'(x) as a fraction. Set denominator of f'(x) = 0.

Approximate the value of f(x₁ + a) if you know the function goes through point (x₁, y₁).

Find slope m = f'(x₁). Then use point slope equation: y - y₁ = m(x − x₁). Evaluate this line for y at x = x₁ + a.

Find the derivative of f(g(x)).

This is the chain rule. You are finding f'(g(x)) * g'(x).

The line y = mx + b is tangent to the graph of f(x) at (x₁, y₁).

Two relationships are true: 1) The function f and the line share the same slope at x₁: m = f'(x₁) 2) The function f and the line share the same y-value at x₁.

Find the derivative of the inverse to f(x) at x = a.

Follow this procedure: 1) Interchange x and y in f(x). 2) Plug the x-value into this equation and solve for y. 3) Using the equation in 1) find dy/dx implicitly. 4) Plug the y-value you found in 2) to dy/dx

Given a piecewise function, show it is differentiable at x = a where the function rule splits.

First, be sure that f(x) is continuous at x = a. Then take the derivative of each piece and show that lim (x→a-) f'(x) = lim (x→a+) f'(x).

Find critical values of f(x).

Find and express f'(x) as a fraction. Set both numerator and denominator equal to zero and solve.

Find the interval(s) where f(x) is increasing/decreasing.

Find critical values of f'(x). Make a sign chart to find the sign of f'(x) in the intervals bounded by critical values. Positive means increasing, negative means decreasing.

Find points of relative extrema of f(x).

Make a sign chart of f'(x). At x = c where the derivative switches from negative to positive, there is a relative minimum. When the derivative switches from positive to negative, there is a relative maximum. To actually find the point, evaluate f(c). OR if f'(c) = 0, then if f''(c) > 0, there is a relative minimum at x = c. If f''(c) < 0, there is a relative maximum at x = c. (2nd Derivative test).

Find inflection points of f(x).

Find and express f''(x) as a fraction. Set both numerator and denominator equal to zero and solve. Make a sign chart of f''(x). Inflection points occur when f''(x) switches from positive to negative or negative to positive.

Find the absolute maximum or minimum of f(x) on [a, b].

Use relative extrema techniques to find relative max/mins. Evaluate f at these values. Then examine f(a) and f(b). The largest of these is the absolute maximum and the smallest of these is the absolute minimum.

Find range of f(x) on (-∞, ∞).

Use relative extrema techniques to find relative max/mins. Evaluate f at these values. Then examine f(a) and f(b). Then examine lim (x→∞) f(x) and lim (x→-∞) f(x).

Find range of f(x) on [a, b].

Use relative extrema techniques to find relative max/mins. Evaluate f at these values. Then examine f(a) and f(b).

Show that Rolle's Theorem holds for f(x) on [a, b].

Show that f is continuous and differentiable on [a, b]. If f(a) = f(b), then find some c on [a, b] such that f'(c) = 0.

Show that the Mean Value Theorem holds for f(x) on [a, b].

Show that f is continuous and differentiable on [a, b]. If f(a) ≠ f(b), then find some c on [a, b] such that f'(c) = f(b)-f(a) / b-a

Given a graph of f'(x), determine intervals where f(x) is increasing/decreasing.

Make a sign chart of f'(x) and determine the intervals where f'(x) is positive and negative.

Determine whether the linear approximation for f(x₁ + a) over-estimates or under-estimates f(x₁ + a).

Find slope m = f'(x₁). Then use point slope equation: y - y₁ = m(x − x₁). Evaluate this line for y at x = x₁ + a. If f''(x) > 0, f is concave up at x₁ and the linear approximation is an underestimation for f(x₁ + a). If f''(x) < 0, f is concave down at x₁ and the linear approximation is an overestimation for f(x₁ + a).

Find intervals where the slope of f(x) is increasing.

Find the derivative of f'(x) which is f''(x). Find critical values of f''(x) and make a sign chart of f''(x) looking for positive intervals.

Find the minimum slope of f(x) on [a, b].

Find the derivative of f'(x) which is f''(x). Find critical values of f''(x) and make a sign chart of f''(x). Values of x where f''(x) switches from negative to positive are potential locations for the minimum slope. Evaluate f'(x) at those values and also f'(a) and f'(b) and choose the least of these values.

Approximate ∫(a to b) f(x) dx using left Riemann sums with n rectangles.

(b-a / n) [f(x₀) + f(x₁) + f(x₂) + … + f(xₙ₋₁)]

Approximate ∫(a to b) f(x) dx using right Riemann sums with n rectangles.

(b-a / n) [f(x₁) + f(x₂) + f(x₃) + … + f(xₙ)]

Approximate ∫(a to b) f(x) dx using midpoint Riemann sums.

Typically done with a table of points. Be sure to use only values that are given. If you are given 7 points, you can only calculate 3 midpoint rectangles.

Approximate ∫(a to b) f(x) dx using trapezoidal summation.

(b-a / 2n) [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(xₙ)]

Find ∫(a to b) f(x) dx where a > b.

-∫(b to a) f(x) dx

Meaning of ∫(a to b) f(t) dt.

The accumulation function - accumulated area under function f starting at some constant a and ending at some variable x.

Given ∫(a to b) f(x) dx, find ∫(a to b) [f(x) + k] dx.

∫(a to b) f(x) dx + ∫(a to b) k dx

Given the value of F(a) where the antiderivative of f is F, find F(b).

Use the fact that ∫(a to b) f(x) dx = F(b) - F(a) so F(b) = F(a) + ∫(a to b) f(x) dx.

Find d/dx ∫(a to x) f(t) dt.

f(x). The 2nd Fundamental Theorem.

Find d/dx ∫(a to g(x)) f(t) dt.

f(g(x)) * g'(x). The 2nd Fundamental Theorem.

Find the area under the curve f(x) on the interval [a, b].

∫(a to b) f(x) dx

Find the area between f(x) and g(x).

Find the intersections, a and b of f(x) and g(x). If f(x) ≥ g(x) on [a, b], then area A = ∫(a to b) [f(x) - g(x)] dx.

Find the line x = c that divides the area under f(x) on [a, b] into two equal areas.

∫(a to c) f(x) dx = ∫(c to b) f(x) dx or ∫(a to c) f(x) dx = 1/2 ∫(a to b) f(x) dx

Find the volume when the area under f(x) is rotated about the x-axis on the interval [a, b].

Disks: Radius = f(x): V = π ∫(a to b) [f(x)]² dx

Find the volume when the area between f(x) and g(x) is rotated about the x-axis.

Establish the interval where f(x) ≥ g(x) and the values of a and b, where f(x) = g(x). V = π ∫(a to b) ([f(x)]² - [g(x)]²) dx

Given a base bounded by f(x) and g(x) on [a, b] the cross sections of the solid perpendicular to the x-axis are squares. Find the volume.

Base = f(x) - g(x). Area = base² = [f(x) - g(x)]². Volume = ∫(a to b) [f(x) - g(x)]² dx

Solve the differential equation dy/dx = f(x)g(y).

Separate the variables: x on one side, y on the other with the dx and dy in the numerators. Then integrate both sides, remembering the +C, usually on the x-side.

Find the average value of f(x) on [a, b].

∫(a to b) f(x) dx / (b-a)

Find the average rate of change of F'(x) on [t₁, t₂].

F'(t₂) - F'(t₁) / t₂ - t₁

y is increasing proportionally to y.

dy/dt = ky which translates to y = Ce^(kt)

Given dy/dx, draw a slope field.

Use the given points and plug them into dy/dx, drawing little lines with the calculated slopes at the points.

Given the position function s(t) of a particle moving along a straight line, find the velocity and acceleration.

v(t) = s'(t), a(t) = v'(t) = s''(t)

Given the velocity function v(t) and s(0), find s(t).

s(t) = ∫ v(t) dt + C. Plug in s(0) to find C.

Given the acceleration function a(t) of a particle at rest and s(0), find s(t).

v(t) = ∫ a(t) dt + C₁. Plug in v(0) = 0 to find C₁. s(t) = ∫ v(t) dt + C₂. Plug in s(0) to find C₂.

Given the velocity function v(t), determine if a particle is speeding up or slowing down at t = k.

Find v(k) and a(k). If both have the same sign, the particle is speeding up. If they have different signs, the particle is slowing down.

Given the position function s(t), find the average velocity on [t₁, t₂].

s(t₂) - s(t₁) / t₂ - t₁

Given the position function s(t), find the instantaneous velocity at t = k.

s'(k)

Given the velocity function v(t) on [t₁, t₂], find the minimum acceleration of a particle.

Find a(t) and set a'(t) = 0. Set up a sign chart and find critical values. Evaluate the acceleration at critical values and also t₁ and t₂ to find the minimum.

Given the velocity function v(t), find the average velocity on [t₁, t₂].

∫(t1 to t2) v(t) dt / (t₂ - t₁)

Given the velocity function v(t), determine the difference of position of a particle on [t₁, t₂].

∫(t1 to t2) v(t) dt

Given the velocity function v(t), determine the distance a particle travels on [t₁, t₂].

Set v(t) = 0 and make a sign chart of v(t) = 0 on [t₁, t₂]. On intervals [a, b] where v(t) > 0, ∫(a to b) |v(t)| dt = ∫(a to b) v(t) dt. On intervals [a, b] where v(t) < 0, ∫(a to b) |v(t)| dt = -∫(a to b) v(t) dt

Given the velocity function v(t) and s(0), find the greatest distance of the particle from the starting position on [0, 4].

Generate a sign chart of v(t) to find turning points. s(t) = ∫ v(t) dt + C. Plug in s(0) to find C. Evaluate s(t) at all turning points and find which one gives the maximum distance from s(0).

The volume of a solid is changing at the rate of …

dV/dt

The meaning of ∫(a to b) R'(t) dt.

This gives the accumulated change of R(t) on [a, b]. ∫(a to b) R'(t) dt = R(b) - R(a) or R(b) = R(a) + ∫(a to b) R'(t) dt

Given a water tank with g gallons initially, filled at the rate of F(t) gallons/min and emptied at the rate of E(t) gallons/min on [t₁, t₂] a) The amount of water in the tank at t = m minutes. b) the rate the water amount is changing at t = m minutes and c) the time t when the water in the tank is at a minimum or maximum

a) g + ∫(t1 to m) [F(t) - E(t)] dt b) d/dt ∫(0 to m) [F(t) - E(t)] dt = F(m) - E(m) c) set F(m) - E(m) = 0, solve for m, and evaluate g + ∫(t1 to m) [F(t) - E(t)] dt at values of m and also the endpoints.