AP Calculus AB – Practice Set

Evaluate each limit.(a) $\displaystyle\lim_{x \to 3} \frac{x^2 – 9}{x – 3}$(b) $\displaystyle\lim_{x \to \infty} \frac{3x^2 + 1}{2x^2 – x}$(c) $\displaystyle\lim_{x \to 0} \frac{\sin 5x}{x}$(d) $\displaystyle\lim_{x \to 0^+} \frac{\ln x}{x}$

Find $\dfrac{dy}{dx}$ for each function.(a) $y = x^2 e^{3x}$(b) $y = \ln(\cos x)$(c) $y = \dfrac{x}{x^2 + 1}$(d) $y = \arctan(2x)$

The curve $x^2 + y^2 = 25$ passes through the point $(3, 4)$.(a) Use implicit differentiation to find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.(b) Write an equation of the line tangent to the curve at $(3, 4)$.

Evaluate each integral.(a) $\displaystyle\int (3x^2 – 4x + 1)\,dx$(b) $\displaystyle\int_0^{\pi/2} \cos x\,dx$(c) $\displaystyle\int \frac{x}{x^2 + 1}\,dx$(d) $\displaystyle\int_1^{e} \frac{\ln x}{x}\,dx$

Let $g(x) = \displaystyle\int_1^{x^2} \sin t\,dt$.(a) Find $g'(x)$.(b) Find $g’\left(\sqrt{\pi/2}\right)$.

Let $f(x) = \dfrac{x^2 – 4}{x – 2}$ for $x e 2$ and $f(2) = k$.(a) Find the value of $k$ that makes $f$ continuous at $x = 2$.(b) With this value of $k$, is $f$ differentiable at $x = 2$? Justify your answer.(c) The function $h(x) = \dfrac{1}{x – 2}$ has a discontinuity at $x = 2$. State the type of discontinuity.

A particle moves along the $x$-axis with velocity $v(t) = t^2 – 4t + 3$ for $0 \le t \le 5$, where $t$ is in seconds and $v$ in metres per second. At $t = 0$ the particle is at $x = 2$.(a) Find all times at which the particle is at rest.(b) Find the displacement of the particle over $0 \le t \le 5$.(c) Find the total distance travelled over $0 \le t \le 5$.(d) Find the position of the particle at $t = 5$.(e) Is the speed of the particle increasing or decreasing at $t = 2$? Justify your answer.

A $10$-foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at a constant rate of $2$ feet per second.(a) Find the rate at which the top of the ladder is sliding down the wall when the bottom is $6$ feet from the wall.(b) Find the rate of change of the area of the triangle formed by the ladder, the wall and the ground at that instant.

An open-top box with a square base is to have a volume of $32$ cubic feet. Let $x$ be the side length of the base and $h$ the height.(a) Express the surface area $S$ of the box (base plus four sides) as a function of $x$ alone.(b) Find the dimensions that minimize the surface area, and justify that your answer gives a minimum.(c) State the minimum surface area.

Consider the differential equation $\dfrac{dy}{dx} = xy$ with $y(1) = 2$.(a) Write an equation of the line tangent to the solution curve at $(1, 2)$ and use it to approximate $y(1.2)$.(b) Find the particular solution $y = f(x)$.(c) Use your solution to find $y(1.2)$ to three decimal places, and state whether the tangent-line approximation is an over- or under-estimate. Justify using $\dfrac{d^2y}{dx^2}$.

Let $R$ be the region in the first quadrant bounded by $y = \sqrt{x}$ and $y = \dfrac{x}{2}$.(a) Find the $x$-coordinates of the points of intersection.(b) Find the area of $R$.(c) Find the volume of the solid generated when $R$ is rotated about the $x$-axis.(d) The region $R$ is the base of a solid whose cross-sections perpendicular to the $x$-axis are squares. Find the volume of this solid.

Let $f(x) = x^3 – 6x^2 + 9x + 1$.(a) Find the intervals on which $f$ is increasing and decreasing.(b) Find the $x$-coordinates of all relative extrema and classify each.(c) Find the point of inflection of the graph of $f$.(d) Find the absolute maximum and absolute minimum values of $f$ on $[0, 4]$.(e) Verify that the Mean Value Theorem applies to $f$ on $[0, 4]$ and find all values of $c$ guaranteed by the theorem.