Definition of the Derivative and Basic Rules: AP Calculus AB Math Test – PDF Download

Question 1: Limit definition (2 points)

Let \(f(x)=x^{2}-3x\). Use the limit definition of the derivative to find \(f^{\prime}(x)\).

Question 2: Basic rules (3 points)

Find the derivative of each function.

1. \(f(x)=5x^{4}-2\sqrt{x}+7\) (1 point)
2. \(g(x)=3e^{x}-4\sin x+2\ln x\) (1 point)
3. \(h(x)=\frac{6}{x^{2}}+3\cos x\) (1 point)

Question 3: Product rule (2 points)

Let \(f(x)=x^{3}e^{x}\). Find \(f^{\prime}(x)\) and evaluate \(f^{\prime}(1)\).

Question 4: Quotient rule (2 points)

Let \(g(x)=\frac{2x+1}{x^{2}+1}\). Find \(g^{\prime}(x)\) and evaluate \(g^{\prime}(0)\).

Question 5: Tangent lines to a cubic (3 points)

The graph of \(f(x)=x^{3}-3x\) is shown.

1. Write an equation of the tangent line at \(x=2\). (1 point)
2. Find every \(x\) where the tangent line is horizontal. (1 point)
3. Find every \(x\) where the tangent line is parallel to \(y=6x\). (1 point)

Question 6: Differentiable piecewise function (2 points)

Let \(f(x)=x^{2}+1\) when \(x\le 2\) and \(f(x)=ax+b\) when \(x > 2\). Find \(a\) and \(b\) so that \(f\) is differentiable at \(x=2\).

Question 7: Estimating rates from data (2 points)

The temperature \(T\) in \(^{\circ}\)F of a cup of coffee is measured \(t\) minutes after it is poured.

1. Find the average rate of change of \(T\) on \([0,4]\). Include units. (1 point)
2. Estimate \(T^{\prime}(6)\) using the points at \(t=4\) and \(t=8\), and explain what it means. (1 point)

Question 8: Derivative with tangent (2 points)

Let \(f(x)=x\tan x\). Find \(f^{\prime}(x)\) and evaluate \(f^{\prime}(\frac{\pi}{4})\).

Question 9: Motion along a line (2 points)

A particle moves along a line with position \(s(t)=t^{3}-6t^{2}+9t\) feet, \(t\ge 0\) seconds.

1. Find the velocity \(v(t)=s^{\prime}(t)\). (1 point)
2. Find every time the particle is at rest, and the position at each. (1 point)

See every chapter on the AP Calculus AB math page.