AP (Advanced Placement) Calculus — practice questions
44 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice AP (Advanced Placement) Calculus in the app →A student differentiates f(x) = x^5 using the power rule. The derivative is:A cost function is the constant f(x) = 7. Its derivative is:For f(x) = 3x^2, a student pulls the 3 out and differentiates the rest. The derivative is:If f(x) = x^3 + x, then f'(x) equals:A function is written as one function multiplied by another. The rule to reach for is the:A student guesses the derivative of a product is simply the product of the derivatives. Is that right?A function appears as one function divided by another. The rule to use is the:A student meets a composition, with one function sitting inside another. The rule for that is the:Differentiating f(x) = (2x + 1)^3 with the chain rule gives:A physics student needs the derivative of sin x. It is:The same student now needs the derivative of cos x. It is:A function is known to be differentiable at a point. What follows about continuity there?Another function is continuous at a point. Must it also be differentiable there?A student differentiates the derivative one more time. The result is the:Third and fourth derivatives, and those beyond, are known together as:Velocity is the rate of change of position. The rate of change of velocity is:An equation cannot be solved neatly for y, so a student differentiates both sides as they stand. That method iKnowing the derivative of a function, a student can also compute the derivative of its:A student differentiates e^x. The result is:The inverse of the natural exponential function has its own name. It is the:A calculus course opens by going back to secant lines and one other kind of line. That other kind is:Average speed over a whole trip differs from speed at one instant. That second quantity is:A student asks whether integration has its own product rule. The honest answer is that there:Instead, a technique built on the differentiation product rule swaps one integral for another. That technique Deciding which factor to call u, a student uses a memory aid ordering logarithmic, inverse, algebraic, trigonoAn integrand holds an algebraic function and a trigonometric one. Under that ordering, u should be the:After finding an antiderivative, a student can confirm it is right by:An integral involving a square root of a quadratic often yields to a technique using sine, tangent or secant. To integrate a rational function, a student splits it into simpler fractions. That method is:The exact form that decomposition takes depends heavily on the factorisation of the:A definite integral was first defined as the limit of sums built on rectangles. Those sums are:Approximating an integral with rectangles whose heights come from the centre of each subinterval uses the:A method that approximates the region with parabolic arcs rather than straight tops is:Comparing an estimate with the true value gives two quantities: relative error and:Before computing, a student wants to know how far off an estimate could be. A theorem supplies error:An integral runs all the way to infinity, or the integrand blows up inside the interval. Such integrals are caOne kind of such integral is taken over an interval that is:The other kind is taken over a closed interval where the function has an infinite:Evaluating such an integral, a student finds the limit does not exist. The integral is then said to:When the limit does exist and gives a finite value, the integral is said to:Some improper integrals cannot be evaluated directly, so a student compares them with a chosen integral. That Integration by parts also has a version adapted for use with:Before choosing a technique, a student must first recognise when integration by parts is:For a rational function, the first thing to recognise is the presence of simple: