Unit 5 Overview (College Board AP® Calculus AB): Exam Questions

Consider the function $y=f\left(x\right)$ whose curve is given by the equation $2y^2-6=y\sin x$ for $y>0$.

Show that $\frac{\text{d}y}{\text{d}x}=\frac{y\cos x}{4y-\sin x}$.

Write an equation for the line tangent to the curve at the point $(0,\sqrt{3})$.

For $0\le x\le \pi$ and $y>0$, find the coordinates of the point where the line tangent to the curve is horizontal.

Determine whether $f$ has a relative minimum, a relative maximum, or neither at the point found in part (c). Justify your answer.

Let $f$ be the function defined by $f\left(x\right)=e^x \text{cos}x$.

Find the average rate of change of $f$ on the interval $0\le x\le \pi$.

What is the slope of the line tangent to the graph of $f$ at $x=\frac{3\pi }{2}$?

Find the absolute minimum value of $f$ on the interval $0\le x\le 2\pi$. Justify your answer.

Let $g$ be a differentiable function such that $g\left(\frac{\pi }{2}\right)=0$. The graph of $g\text{'}$, the derivative of $g$, is shown below.

Find the value of $\underset{x\to \pi /2}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$, or state that it does not exist. Justify your answer.

Let $f$ be the function defined by $f\left(x\right)=\text{cos}\left(2x\right)+e^{\text{sin}x}$. Let $g$ be a differentiable function. The table below gives values of $g$ and its derivative $g\text{'}$ at selected values of $x$.

Let $h$ be the function whose graph, consisting of five line segments, is shown below.

Find the slope of the line tangent to the graph of $f$ at $x=\pi$.

Let $k$ be the function defined by $k\left(x\right)=h\left(f\right(x\left)\right)$. Find $k\text{'}\left(\pi \right)$.

Let $m$ be the function defined by $m\left(x\right)=g(-2x) h\left(x\right)$. Find $m\text{'}\left(2\right)$.

Is there a number $c$ in the closed interval $[-5,-3]$ such that $g\text{'}\left(c\right)=-4$? Justify your answer.