Calculus 1st Semester Review

Front

What is the derivative of A = 2x^3 with respect to time t?

Back

dA/dt = 6x^2 (dx/dt)

Front

If the radius of an oil spill increases at a rate of 5 m/min, how do you find the rate at which the area of the spill is increasing?

Back

Area A = πr^2; dA/dt = 2πr(dr/dt). When r = 5 m, dA/dt = 50π m²/min.

Front

What does it mean if f''(x) > 0 over an interval?

Back

It means that f'(x) is increasing over that interval.

Front

Find the second derivative of f(x) = x^2 + e^x - cos(x).

Back

f''(x) = 2 + e^x + sin(x)

Front

At what point does a bug change direction if its velocity function is given?

Back

The bug changes direction at t = 6.5.

Front

What is the chain rule in calculus?

Back

If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

Define the concept of limits in calculus.

Back

A limit is the value that a function approaches as the input approaches a certain point.

Front

What is the Fundamental Theorem of Calculus?

Back

It links the concept of differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫_a^b f(x)dx = F(b) - F(a).

Front

What is the product rule for differentiation?

Back

If u and v are functions of x, then d(uv)/dx = u'v + uv'.

Front

What is the quotient rule for differentiation?

Back

If u and v are functions of x, then d(u/v)/dx = (u'v - uv')/v^2.