AP EAMCET / TS EAMCET Mathematics - Calculus, Geometry and Probability Syllabus

How do you find a unit vector in the direction of vector a?

a_hat = a / |a|, where |a| is the magnitude of a.

Give the position vector of the point dividing the line joining vectors a and b in ratio m:n internally.

r = (m*b + n*a)/(m + n)

Define the scalar (dot) product of two vectors and give its formula.

a . b = |a||b| cos theta, where theta is the angle between them. In components: a.b = a1*b1 + a2*b2 + a3*b3.

What is the dot-product condition for two non-zero vectors to be perpendicular?

a . b = 0

Define the vector (cross) product and give its magnitude.

a x b is a vector perpendicular to both a and b with magnitude |a||b| sin theta, directed by the right-hand rule. |a x b| equals the area of the parallelogram with sides a and b.

How is the cross product a x b computed using a determinant?

a x b = |i j k; a1 a2 a3; b1 b2 b3| = (a2*b3 - a3*b2)i - (a1*b3 - a3*b1)j + (a1*b2 - a2*b1)k.

What is the scalar triple product [a b c] and what does it represent geometrically?

[a b c] = a . (b x c); its absolute value equals the volume of the parallelepiped with edges a, b, c. It is zero if the three vectors are coplanar.

State the vector triple product expansion of a x (b x c).

a x (b x c) = (a . c)b - (a . b)c

Define the limit of a function and state the condition for a limit to exist at x = a.

lim(x->a) f(x) = L means f(x) approaches L as x approaches a. The limit exists iff the left-hand limit equals the right-hand limit.

State two standard limits: lim(x->0) sin(x)/x and lim(x->0) (e^x - 1)/x.

lim(x->0) sin(x)/x = 1; lim(x->0) (e^x - 1)/x = 1.

What three conditions must hold for a function f to be continuous at x = a?

(1) f(a) is defined, (2) lim(x->a) f(x) exists, (3) lim(x->a) f(x) = f(a).

State the first principle (definition) of the derivative of f(x).

f'(x) = lim(h->0) [f(x+h) - f(x)] / h

State the product rule and quotient rule for differentiation.

Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v - uv')/v^2.

State the chain rule for differentiating y = f(g(x)).

dy/dx = f'(g(x)) * g'(x), i.e., dy/dx = (dy/du)(du/dx).

Give the derivatives of sin x, cos x, tan x, and ln x.

d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(tan x) = sec^2 x; d/dx(ln x) = 1/x.

How do you find the maxima and minima of a function using the second derivative test?

At a stationary point where f'(x) = 0: if f''(x) < 0 it is a local maximum, if f''(x) > 0 it is a local minimum, and if f''(x) = 0 the test is inconclusive.

What is the geometric meaning of the derivative f'(x) at a point?

It is the slope of the tangent to the curve y = f(x) at that point; the slope of the normal is -1/f'(x).

State Rolle's Theorem.

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) such that f'(c) = 0.

State the Lagrange Mean Value Theorem.

If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Give the integrals of x^n and 1/x.

Integral of x^n dx = x^(n+1)/(n+1) + C (n != -1); Integral of (1/x) dx = ln|x| + C.

State the formula for integration by parts.

Integral of u dv = u*v - Integral of v du. Choose u by ILATE order (Inverse, Log, Algebraic, Trig, Exponential).

Give the integrals of sin x, cos x, and sec^2 x.

Integral of sin x dx = -cos x + C; Integral of cos x dx = sin x + C; Integral of sec^2 x dx = tan x + C.

State the Fundamental Theorem of Calculus for definite integrals.

If F is an antiderivative of f, then the integral from a to b of f(x) dx = F(b) - F(a).

State the property: integral from a to b of f(x) dx equals what in terms of f(a+b-x)?

Integral from a to b of f(x) dx = Integral from a to b of f(a + b - x) dx.

When is the integral from -a to a of f(x) dx equal to 2 times integral from 0 to a, and when is it zero?

It equals 2 * integral from 0 to a of f(x) dx if f is even (f(-x) = f(x)), and equals 0 if f is odd (f(-x) = -f(x)).

Define the order and degree of a differential equation.

Order is the order of the highest derivative present; degree is the power of the highest-order derivative after the equation is made polynomial in derivatives (free of radicals/fractions).

How do you solve a first-order linear differential equation dy/dx + Py = Q?

Find integrating factor IF = e^(Integral P dx); the solution is y * IF = Integral (Q * IF) dx + C.

How is a variable-separable differential equation solved?

Rewrite as f(y) dy = g(x) dx, then integrate both sides: Integral f(y) dy = Integral g(x) dx + C.

Define mean deviation, variance, and standard deviation as measures of dispersion.

Mean deviation = mean of absolute deviations from a central value; Variance = mean of squared deviations from the mean; Standard deviation = sqrt(variance).

State the addition theorem of probability for two events A and B.

P(A union B) = P(A) + P(B) - P(A intersection B). For mutually exclusive events, P(A union B) = P(A) + P(B).

State the conditional probability formula and the multiplication theorem.

P(A|B) = P(A intersection B)/P(B). Multiplication: P(A intersection B) = P(A)*P(B|A) = P(B)*P(A|B). For independent events P(A intersection B) = P(A)P(B).

State Bayes' Theorem for events A_i with prior probabilities.

P(A_i|B) = [P(A_i)*P(B|A_i)] / sum over j of [P(A_j)*P(B|A_j)].

For a binomial distribution with parameters n and p, give P(X = r), the mean, and the variance.

P(X = r) = C(n, r) p^r q^(n-r) with q = 1 - p; mean = np; variance = npq.

For a Poisson distribution with parameter lambda, give P(X = r), the mean, and the variance.

P(X = r) = (e^(-lambda) * lambda^r)/r!; mean = lambda; variance = lambda.

What is the expected value (mean) and variance of a discrete random variable X?

Mean E(X) = sum of x_i * P(x_i); Variance Var(X) = E(X^2) - [E(X)]^2 = sum x_i^2 P(x_i) - (E(X))^2.