Comprehensive Calculus Non-Calculator and Calculator Exam Guide

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Core Themes

The exam is centered around two primary topics:

• Derivatives: Represents approximately $85\%$ of the assessment across various representations, notations, and techniques.

• Tangent Lines: Finding equations of tangent lines and using them for local linear approximation.

Non-Calculator Section

1. Derivative Representations and Techniques

   • Notation Diversity: Derivatives appear in multiple forms, including $f'(x)$, $y'$, $\frac{d}{dx}$, and $\frac{dy}{dx}$.

   • Composite and Symbolic Rules: Applying the chain rule to composite functions such as $f(g(x))$ without explicit formulas.

   • Implicit Differentiation: Differentiating expressions where $x$ and $y$ are implicitly defined.

   • Higher-Order Derivatives: Progressing from a given first derivative $\frac{dy}{dx}$ to the third derivative $\frac{d^3y}{dx^3}$ by differentiating twice more.

   • Trigonometric Values: Evaluating derivatives at points requiring knowledge of the unit circle (e.g., at $\sqrt{2\pi}$).

2. Graphical Derivative Problems

   • Extracting function values (vertical position) and derivative values (slopes) from provided graphs.

   • Using product rules on combinations such as $h(x) = f(x) \cdot g(x)$ or $h(x) = e^x \cdot g(x)$ to calculate $h'(x)$.

3. Tangent Lines and Approximations

   • Finding slope using the derivative evaluated at a point: $m = f'(x_0)$.

   • Writing linear tangent line equations using point-slope form.

   • Using tangent line equations to approximate values (e.g., approximating at $x = 1.5$).

4. Non-Calculator Free Response (FRQ)

   • Show That / Verification: Showing explicit algebraic and calculus steps from an equation to a given derivative $\frac{dy}{dx}$.

   • Setting Derivatives Equal to a Value: Setting $\frac{dy}{dx} = 1$ and solving analytically using unit circle values.

   • Tangent Line Approximation: Using a given point and derivative to construct a tangent line and evaluate an approximation at $x = 1.8$.

Calculator-Active Section

1. Numerical and Table-Based Analysis

   • Average Rate of Change: Calculating $\frac{f(b) - f(a)}{b - a}$ over an interval such as $[0, 10]$.

   • Table Problems: Differentiating combinations of functions using values and slopes selected from tables of $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$.

2. Calculator Utility and Graphs

   • Finding zeros and calculating numerical derivatives at specific points using built-in calculator functions.

   • Identifying values where two functions have parallel tangent lines by equating their derivative values ($f'(x) = g'(x)$).

   • Comparing the relative magnitudes of slopes and function values from graphs.

3. Calculator Free Response (FRQ)

   • Contextual Interpretation: Estimating derivatives (e.g., estimating $W'(12)$ from table data) and interpreting the result as a rate of change within the context of a real-world scenario (e.g., water temperature change over time).

   • Equation of Tangent Line: Finding analytic derivatives and writing tangent line equations.

Non-Calculator Section

1. Question 1 (Composite Symbolic Derivative): Apply the chain rule to evaluate the derivative of $f(g(x))$ where $f(x)$ is defined analytically and $g(x)$ is symbolic.

2. Question 2 (Trigonometric Evaluation): Evaluate the derivative $f'(\sqrt{2\pi})$ analytically using unit circle trigonometric values.

3. Question 3 (Implicit Differentiation): Compute $\frac{dy}{dx}$ from an equation where $x$ and $y$ are implicitly defined.

4. Question 4 (Slope of Tangent Line): Differentiate a given equation $y = f(x)$ and evaluate it at a given point to determine the slope of the tangent line.

5. Question 5 (Derivative Evaluation): Differentiate a function $f(x)$ and compute the value of $f'(2)$.

6. Question 6 (Product Rule from Graph): Use graphs of $f(x)$ and $g(x)$ to determine $h'(2)$ for $h(x) = f(x) \cdot g(x)$, using heights for function values and slopes for derivatives.

7. Question 7 (Derivative Notation): Compute $\frac{d}{dx}[f(x)]$ using analytical differentiation rules.

8. Question 8 (Equation of Tangent Line): Compute the slope from the derivative at a point and write the full equation of the tangent line.

9. Question 9 (Derivative Formula): Differentiate a given function $f(x)$ to determine its derivative expression $f'(x)$.

10. Question 10 (Tangent Line to Implicit Curve): Identify the correct equation of the tangent line to an implicit curve at a specified point.

11. Question 11 (Higher-Order Derivative): Start from a given first derivative $\frac{dy}{dx}$ and differentiate twice more to calculate the third derivative $\frac{d^3y}{dx^3}$.

12. Question 12 (Derivative Evaluation at a Value): Find $\frac{dy}{dx}$ for a given curve and evaluate it at $x = 3$.

13. Question 13 (Derivative Evaluation): Differentiate a given function $f(x)$ and evaluate $f'(2)$.

14. Question 14 (Product Rule with Exponential from Graph): Calculate $h'(-1)$ for $h(x) = e^x \cdot g(x)$ using values and tangent line slopes from a provided graph of $g(x)$.

15. Question 15 (Derivative Evaluation): Compute the derivative value $f'(1)$ for a given function $f(x)$.

16. Question 16 (Tangent Line Approximation): Construct the equation of a tangent line at a given point and use it to approximate a function value at $x = 1.5$.

17. Non-Calculator Free Response (FRQ):

    • Part A: Show all intermediate algebraic and differentiation steps to verify that an equation yields a specified derivative $\frac{dy}{dx}$.

    • Part B: Set the provided derivative $\frac{dy}{dx} = 1$ and solve for $x$ using the unit circle.

    • Part C: Construct the equation of a tangent line using a given point and derivative, then approximate the value at $x = 1.8$.

Calculator-Active Section

1. Question 18 (Average Rate of Change): Compute the average rate of change $\frac{f(b) - f(a)}{b - a}$ over the interval $[0, 10]$ using a table of values.

2. Question 19 (Table Problem with Rules): Find $h'(0)$ for a combined function (such as $h(x) = 4f(x) \cdot g(x)$) using function values and slopes from a table.

3. Question 20 (Parallel Tangent Lines): Determine the point where the tangent line of $g(x)$ is parallel to the tangent line of $f(x)$ by solving $f'(x) = g'(x)$.

4. Question 21 (Numerical Derivative at an Intercept): Use built-in calculator functions to find the $x$-intercept of a function and evaluate its derivative at that zero.

5. Question 22 (Comparing Function and Slope Values): Analyze a graph to compare the relative magnitudes of function values and derivative slopes (such as evaluating if $f'(0) < f(1)$).

6. Question 23 (Table Problem): Apply differentiation rules to find $h'(1)$ using tabulated values of $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$.

7. Question 24 (Comparing Slopes): Order the magnitudes of derivative slopes across multiple points on a graph (such as comparing $f'(2)$, $f'(0)$, and $f'(1)$).

8. Question 25 (Inverse Trigonometric Derivative): Compute the derivative of an expression involving $\arcsin(x)$.

9. Calculator Free Response (FRQ):

   • Part A: Estimate $W'(12)$ from a table and interpret the result in the context of the problem as a rate of change.

   • Part B: Analytically differentiate a given function $f(x)$ to find $f'(x)$.

   • Part C: Write the equation of the tangent line to $f(x)$ at a specified point.