Calculus Exam Info

• Statement of properties, e.g., limit of a constant, sum, product or quotient
• Limit calculations, including limits involving infinity, e.g., $\underset{x\to 0}{\lim }\frac{\sin \left(x\right)}{x}=1$, $\underset{x\to 0}{\lim }\frac{1}{x}$ is nonexistent, and $\underset{x\to \infty }{\lim }\frac{\sin \left(x\right)}{x}=0$
• Continuity

The Derivative

• Definitions of the derivative, e.g., $f\prime \left(a\right)=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ and $f\prime \left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$
• Derivatives of elementary functions
• Derivatives of sums, products and quotients (including $\mathit{t}\mathit{a}\mathit{n}\mathbf{ }\mathit{x}$ and $\mathit{c}\mathit{o}\mathit{t}\mathbf{ }\mathit{x}$
• Derivative of a composite function (chain rule), e.g., $\mathit{s}\mathit{i}\mathit{n}(ax+b)$, $\mathit{a}{\mathit{e}}^{\mathbf{k}\mathbf{x}}$, $\mathit{l}\mathit{n}\left(kx\right)$
• Implicit differentiation
• Derivative of the inverse of a function (including $\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{x}$ and $\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\mathbf{ }\mathit{x}$)
• Higher order derivatives
• Corresponding characteristics of graphs of $\mathit{f}$, $\mathit{f}\mathbf{\text{'}}$ and $\mathit{f}\mathbf{\text{'}\text{'}}$
• Statement of the Mean Value Theorem; applications and graphical illustrations
• Relation between differentiability and continuity
• Use of L'Hospital's Rule (quotient and indeterminate forms)

Applications of the Derivative

• Slope of a curve at a point
• Tangent lines and linear approximation
• Curve sketching: increasing and decreasing functions; relative and absolute maximum and minimum points; concavity; points of inflection
• Extreme value problems
• Velocity and acceleration of a particle moving along a line
• Average and instantaneous rates of change
• Related rates of change

Antiderivatives and Techniques of Integration

• Concept of antiderivatives
• Basic integration formulas
• Integration by substitution (use of identities and change of variable)

Applications of Antiderivatives

• Distance and velocity from acceleration with initial conditions
• Solutions of $\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{k}\mathit{y}$ and applications to growth and decay

The Definite Integral

• Definition of the definite integral as the limit of a sequence of Riemann sums and approximations of the definite integral using areas of rectangles
• Properties of the definite integral

• The Fundamental Theorem: $\frac{d}{dx}{\int }_a^{x}f\left(t\right)dt=f\left(x\right)$ and ${\int }_a^{b}F\text{'}\left(x\right)dx=F\left(b\right)-F\left(a\right)$

   

Applications of the Definite Integral

• Average value of a function on an interval
• Area, including area between curves
• Other (e.g., accumulated change from a rate of change)

Notes and Reference Information

1. Figures that accompany questions are intended to provide information useful in answering the questions. All figures lie in a plane unless otherwise indicated. The figures are drawn as accurately as possible except when it is stated in a specific question that the figure is not drawn to scale. Straight lines and smooth curves may appear slightly jagged.
2. Unless otherwise specified, all angles are measured in radians, and all numbers used are real numbers.
3. Unless otherwise specified, the domain of any function $f$ is assumed to be the set of all real numbers $x$ for which $f\left(x\right)$ is a real number. The range of $f$ is assumed to be the set of all real numbers $f\left(x\right)$ where $x$ is in the domain of $f$.
4. In this test, $ln x$ denotes the natural logarithm of $x$ (that is, the logarithm to the base $e$).
5. The inverse of a trigonometric function $f$ may be indicated using the inverse function notation $f^{-1}$ or with the prefix "arc" (e.g., ${\sin }^{-1} x=arc\sin x$)