Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = ln(3te^(-t))

In Exercises 27–32, find dy/dx.

e^(2x)=sin(x+3y)

In Exercises 1–4, solve for t.

e^(sqrt(t)) = x^2

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

19. Show that e^x grows faster as x→∞ than x^n for any positive integer n, even x^1,000,000. (Hint: What is the nth derivative of x^n?)

Evaluate the integrals in Exercises 111–114.

113. ∫₁^(1/x) (1 / t) dt,x > 0