Ehime Entrance Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Derivative as a limit

The derivative at a is f'(a) = lim_{h→0} [f(a+h) − f(a)]/h, provided the limit exists; it equals the slope of the tangent line at x = a.

Tangent line equation

At x = a the tangent to y = f(x) is y − f(a) = f'(a)(x − a). Use it when the point of tangency and the derivative there are known.

First derivative test

If f' changes from positive to negative at c, f has a local maximum there; from negative to positive, a local minimum. f'(c)=0 alone does not guarantee an extremum.

Global extrema on closed interval

For a continuous f on [a,b], evaluate f at all critical points in (a,b) and at both endpoints; the largest and smallest values are the global maximum and minimum.

Fundamental theorem of calculus

If F is an antiderivative of continuous f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a). This links definite integrals with antiderivatives.

Integration by parts

∫ u dv = uv − ∫ v du, and for definite integrals ∫_a^b u v' dx = [uv]_a^b − ∫_a^b u'v dx. Choose u so that u' simplifies.

Area between two curves

If f(x) ≥ g(x) on [a,b], the area between the curves is ∫_a^b [f(x) − g(x)] dx. If they cross, split the interval where the order changes.

Volume by disks

Revolving y=f(x)≥0 about the x-axis over [a,b] gives V = π∫_a^b [f(x)]^2 dx. For a region between two radii use washers.

Arithmetic sequence sum

For an arithmetic sequence with first term a and common difference d, the nth term is a+(n−1)d and the sum of the first n terms is n[2a+(n−1)d]/2.

Geometric series sum

For a geometric sequence with first term a and ratio r≠1, the first-n sum is a(1−r^n)/(1−r). If |r|<1, the infinite sum is a/(1−r).

Mathematical induction

To prove a statement for all positive integers n: verify the base case, assume it for n=k, and use that assumption to prove it for n=k+1.

Squeeze theorem for sequences

If a_n ≤ b_n ≤ c_n eventually and lim a_n = lim c_n = L, then lim b_n = L. Useful when a sequence oscillates inside bounds.

Dot product formula

For vectors a and b, a·b = |a||b|cosθ = a1b1+a2b2 (plus a3b3 in space). It is zero exactly when nonzero a and b are perpendicular.

Cross product magnitude

In space, |a×b| = |a||b|sinθ equals the area of the parallelogram spanned by a and b; half of it is the area of the triangle.

Equation of a plane

A plane through point P with normal vector n satisfies n·(x−p)=0, equivalently n·x = n·p. In coordinates this gives ax+by+cz=d.

Distance point to plane

The distance from point P to plane n·x=d is |n·p−d|/|n|. This follows by projecting the vector from the plane to P onto the normal.

Logarithm laws

For a,b>0 and base c>0, c≠1: log_c(ab)=log_c a+log_c b, log_c(a/b)=log_c a−log_c b, and log_c(a^k)=k log_c a. The domain requires positive arguments.

General sine solution

If sin x = s with s in [−1,1] and α a solution, then x = α+2kπ or x = π−α+2kπ for integers k. Cosine and tangent have analogous forms.

Related rates procedure

Differentiate the geometric relation with respect to time t using the chain rule, substitute known values and rates, then solve for the unknown rate. Keep units consistent.

Exponential derivative

d/dx e^x = e^x, and d/dx a^x = a^x ln a for a>0. For logarithms, d/dx ln x = 1/x for x>0.

Combinations and permutations

nPr = n!/(n−r)! counts ordered selections of r from n distinct objects; nCr = n!/[r!(n−r)!] counts unordered selections. nCr = nC(n−r).

Conditional probability

P(A|B) = P(A∩B)/P(B), provided P(B)>0. Rearranging gives the multiplication rule P(A∩B)=P(A|B)P(B).

Expected value formula

For a discrete random variable X taking values x_i with probabilities p_i, E[X]=Σ x_i p_i. Also E[aX+b]=aE[X]+b.

Polar form multiplication

For z1=r1(cosθ1+i sinθ1) and z2=r2(cosθ2+i sinθ2), the product has modulus r1r2 and argument θ1+θ2; division divides moduli and subtracts arguments.

De Moivre's theorem

[r(cosθ+i sinθ)]^n = r^n(cos nθ+i sin nθ) for integer n. It gives powers quickly and, with r=1, roots of unity.

Complex circle locus

The locus |z−a|=r is the circle centered at a with radius r. The locus |z−a|=|z−b| is the perpendicular bisector of the segment joining a and b.

Remainder theorem

When polynomial P(x) is divided by x−a, the remainder is P(a). In particular, x−a is a factor of P exactly when P(a)=0.

Rational root theorem

For a polynomial with integer coefficients, any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

Euclidean algorithm

For integers a>b>0, repeatedly replace (a,b) by (b, a mod b); the last nonzero remainder is gcd(a,b). This also yields integers x,y with ax+by=gcd(a,b).

Law of cosines

In a triangle with sides a,b,c and angle C opposite side c, c^2=a^2+b^2−2ab cos C. It reduces to Pythagoras when C=90°.

Power of a point

For a point P and a circle, if a line through P meets the circle at A,B, then PA·PB is constant; for a tangent PT, PT^2=PA·PB.

Inscribed angle theorem

An inscribed angle is half the central angle subtending the same arc; angles in the same segment are equal, and an angle in a semicircle is 90°.