Practice Arena

Let f(x) = x / (1 + |x|), where f: R → R. Determine if f is injective, surjective, or bijective.

Let f: N → N be a function defined by f(x) = x² + x + 1. Is f injective?

Let A = R - {3} and B = R - {1}. Consider the function f: A → B defined by f(x) = (x-2)/(x-3). Is f bijective?

Let f(x) = x^2 and g(x) = sin(x). Is gof injective?

Find the domain of the function f(x) = sqrt(log_e( (5x-x^2)/4 )).

Let f(x) = sin(x) + cos(x). Is this function periodic? If so, what is its period?

Find the inverse of the function f(x) = (e^x - e^-x) / (e^x + e^-x) + 2.

Check if the relation R on the set of integers Z defined by (a, b) ∈ R if |a - b| ≤ 1 is an equivalence relation.

The number of onto functions from a set A with 4 elements to a set B with 3 elements is:

Let f(x) = [x] and g(x) = sin(πx), where [.] denotes the greatest integer function. Discuss the continuity of gof(x).

Evaluate lim (x→0) (e^x - 1 - x) / x²

Find the value of k for which the function f(x) = {(sin(5x)/3x, if x≠0), (k, if x=0)} is continuous at x=0.

Evaluate lim (x→∞) x * tan(1/x).

Discuss the continuity of f(x) = x - [x] at integer points, where [x] is the greatest integer function.

Evaluate lim (n→∞) (1/n) * Σ[k=1 to n] (k/n).

Find the constants a and b such that the function f(x) defined by f(x) = {5 if x ≤ 2; ax+b if 2 < x < 10; 21 if x ≥ 10} is continuous.

Evaluate lim (x→0) (cos(x))^(1/x²).

If f(x+y) = f(x) + f(y) for all x,y and f is continuous at x=0, show that f is continuous for all x.

Evaluate lim(x→1) (x + x² + ... + xⁿ - n)/(x-1).

Find the number of points where the function f(x) = [x] + [x + 1/2] is discontinuous in the interval (0, 2). ([.] is G.I.F)

Find the derivative of y = (ln x)^cos(x).

If x = a(cos(t) + t*sin(t)) and y = a(sin(t) - t*cos(t)), find d²y/dx².

If f is a differentiable function such that f(x+y) = f(x)f(y) for all x,y and f(1)=e, f'(0)=1, find f(x).

Find the derivative of f(x) = tan⁻¹((√(1+x²) - 1)/x).

If y = (x + √(1+x²))ⁿ, prove that (1+x²)y₂ + xy₁ - n²y = 0.

If f(x) is a polynomial of degree n, prove that f(x) = f(a) + (x-a)f'(a) + ((x-a)²/2!)f''(a) + ... + ((x-a)ⁿ/n!)f⁽ⁿ⁾(a).

Find the derivative of sin⁻¹(2x√(1-x²)) with respect to sec⁻¹(1/√(1-x²)).

If y = e^(a*sin⁻¹(x)), show that (1-x²)y₂ - xy₁ - a²y = 0.

Find the nth derivative of y = sin(ax+b).

Let g(x) be the inverse of a function f(x). Find g''(x) in terms of f' and f''.

Find the equation of the tangent to the curve y = √(3x-2) which is parallel to the line 4x - 2y + 5 = 0.

A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter is 10m, find the dimensions for maximum light (maximum area).

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Find the intervals in which the function f(x) = (4sin(x) - 2x - x*cos(x)) / (2+cos(x)) is increasing or decreasing.

Verify Lagrange's Mean Value Theorem for f(x) = x(x-1)(x-2) in the interval [0, 1/2].

Find the maximum slope of the curve y = -x³ + 3x² + 9x - 27.

Find the point on the curve y = x³ where the slope of the tangent is equal to the y-coordinate of the point.

Show that for all x > 0, x / (1+x²) < tan⁻¹(x) < x.

Find the condition for the curves ax²+by²=1 and a₁x²+b₁y²=1 to intersect orthogonally.

A point P is on the parabola y² = 4ax. The normal at P cuts the x-axis at N. The locus of the midpoint of PN is:

Evaluate the integral: ∫ e^x * (1 + sin x) / (1 + cos x) dx

Evaluate ∫ dx / (x(x⁵ + 1)).

Evaluate ∫ (x² + 1) / (x⁴ + 1) dx.

Evaluate ∫ √(a² - x²) dx.

Evaluate ∫ dx / (sin(x) + sec(x)).

Evaluate ∫ (x*e^x) / (1+x)² dx.

Evaluate ∫ dx / (x²(x⁴+1)^(3/4)).

Evaluate ∫ (cos(2x) - cos(2a)) / (cos(x) - cos(a)) dx.

Evaluate ∫ dx / (cos(x-a)cos(x-b)).

Evaluate ∫ (sin(x-a)) / sin(x+a) dx.

Evaluate ∫[0, π/2] (sin⁴x) / (sin⁴x + cos⁴x) dx.

Evaluate ∫[0, π] x*log(sin(x)) dx.

Evaluate ∫[-1, 3/2] |x*sin(πx)| dx.

Find the area bounded by the curves y = x*e^|x| and y = x/e^|x|.

Evaluate lim(n→∞) [(n!)^(1/n)] / n.

The area of the region bounded by the curve y=e^x, y=e^-x, and the line x=1 is:

Evaluate ∫[0, 2π] e^cos(x) / (e^cos(x) + e^-cos(x)) dx.

Find the area of the region {(x,y): y² ≤ 2x and x² + y² ≤ 8}.

Evaluate ∫[0,1] cot⁻¹(1 - x + x²) dx.

The value of the integral ∫[0,∞] (ln x)/(1+x²) dx is:

Solve the linear differential equation: (1 + x²)dy + 2xy dx = cot(x) dx

Find the particular solution of the differential equation (x - y)(dx + dy) = dx - dy, given y = -1 when x = 0.

Solve the differential equation dy/dx = (x+y+1)².

Solve y dx - x dy + log(x) dx = 0.

The population of a town grows at a rate proportional to the population. If the population was 400 in 2000 and 1600 in 2020, what is the predicted population for 2040?

Solve dy/dx = y/x + tan(y/x).

A body cools from 80°C to 50°C in 20 minutes in a room at 20°C. Find the time it takes to cool from 80°C to 60°C.

Solve the Bernoulli equation: dy/dx + y/x = x²y⁴.

Find the orthogonal trajectories of the family of parabolas y²=4ax.

Solve the exact differential equation (2xy - sec²x)dx + (x² + 2y)dy = 0.