Applications of Calculus Exam 1 Review – Step-by-Step Guidance

Q1(a). Find the following values: f(-3), f(-1), f(1), f(2), f(3) using the graph of y = f(x).

Background

Topic: Reading Function Values from a Graph

This question tests your ability to interpret a graph and determine the value of a function at specific points.

Key Terms:

• Function value: The y-coordinate of the graph at a given x-value.

Step-by-Step Guidance

1. Locate each x-value on the horizontal axis of the graph: $x = -3, -1, 1, 2, 3$.

2. For each x-value, find the corresponding point on the graph and read the y-coordinate. This is $f(x)$.

3. If the graph does not pass through a point at a given x-value, check for open or closed circles to determine if the function is defined there.

4. If there is a hole (open circle) at a point, the function is undefined at that x-value.

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Q1(b). Find the following limits using the graph: $\lim\limits_{x \to -1^-} f(x)$, $\lim\limits_{x \to -1^+} f(x)$, $\lim\limits_{x \to -1} f(x)$, $\lim\limits_{x \to 1} f(x)$, $\lim\limits_{x \to 2} f(x)$, $\lim\limits_{x \to \infty} f(x)$.

Background

Topic: Limits from Graphs

This question tests your understanding of one-sided and two-sided limits, as well as limits at infinity, using a graph.

Key Terms:

• Left-hand limit: $\lim\limits_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left.

• Right-hand limit: $\lim\limits_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right.

• Two-sided limit: $\lim\limits_{x \to a} f(x)$ exists if both one-sided limits exist and are equal.

• Limit at infinity: $\lim\limits_{x \to \infty} f(x)$ is the value $f(x)$ approaches as $x$ increases without bound.

Step-by-Step Guidance

1. For each limit, trace the graph as $x$ approaches the specified value from the left and right.

2. Observe the y-values the graph approaches from each side. For two-sided limits, check if both sides approach the same value.

3. If the left and right limits are not equal, the two-sided limit does not exist (DNE).

4. For $x \to \infty$, observe the end behavior of the graph as $x$ increases.

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Q1(c). Is $f$ continuous at $x = -1$? Justify.

Background

Topic: Continuity from a Graph

This question tests your understanding of the definition of continuity at a point using graphical information.

Key Terms:

• Continuity at $x = a$: $f$ is continuous at $x = a$ if $\lim\limits_{x \to a} f(x) = f(a)$ and the limit exists.

Step-by-Step Guidance

1. Check if $f(-1)$ is defined (i.e., there is a point on the graph at $x = -1$).

2. Find $\lim\limits_{x \to -1} f(x)$ using the graph (see part (b)).

3. Compare $f(-1)$ and $\lim\limits_{x \to -1} f(x)$. If they are equal and the limit exists, $f$ is continuous at $x = -1$.

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Q1(d). Is $f$ continuous at $x = 1$? Justify.

Background

Topic: Continuity from a Graph

This question tests your ability to apply the definition of continuity at a point.

Key Terms:

• Same as above: $f$ is continuous at $x = a$ if $\lim\limits_{x \to a} f(x) = f(a)$ and the limit exists.

Step-by-Step Guidance

1. Check if $f(1)$ is defined (look for a point at $x = 1$ on the graph).

2. Find $\lim\limits_{x \to 1} f(x)$ using the graph.

3. Compare $f(1)$ and $\lim\limits_{x \to 1} f(x)$. If they are equal and the limit exists, $f$ is continuous at $x = 1$.

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Q2. Graph the piecewise function $f(x) = \begin{cases} x^2 + 1, & x < 0 \\ x - 2, & x \geq 0 \end{cases}$. Find all values of $x$ where $f$ is discontinuous and find the left and right limits at any discontinuity.

Background

Topic: Piecewise Functions and Continuity

This question tests your ability to analyze piecewise-defined functions for points of discontinuity and to compute one-sided limits at those points.

Key Terms and Formulas:

• Piecewise function: A function defined by different expressions on different intervals.

• Discontinuity: A point where the function is not continuous.

• One-sided limits: $\lim\limits_{x \to a^-} f(x)$ and $\lim\limits_{x \to a^+} f(x)$.

Step-by-Step Guidance

1. Identify the point where the formula for $f(x)$ changes (at $x = 0$).

2. Check for continuity at $x = 0$ by finding $\lim\limits_{x \to 0^-} f(x)$, $\lim\limits_{x \to 0^+} f(x)$, and $f(0)$.

3. If the left and right limits are not equal, $f$ is discontinuous at $x = 0$.

4. State the values of the left and right limits at $x = 0$.

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Q3. Use the rules for limits to find the following limits. Use proper limit notation.

Background

Topic: Limit Laws and Algebraic Manipulation

This question tests your ability to apply limit laws and algebraic techniques to evaluate limits.

Key Terms and Formulas:

• Limit laws: Properties that allow you to break up limits over sums, products, quotients, etc.

• Factoring: Useful for simplifying expressions before taking limits.

Step-by-Step Guidance

1. For each limit, substitute the value of $x$ directly if possible.

2. If direct substitution gives an indeterminate form (like $0/0$), factor or simplify the expression.

3. Apply limit laws to evaluate the limit step by step.

4. For limits as $x \to \infty$, analyze the dominant terms in the expression.

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Q4(a). Graph $f(x) = e^{x} - 2$.

Background

Topic: Exponential Functions and Graphing

This question tests your understanding of the graph of exponential functions and vertical shifts.

Key Terms:

• Exponential function: $e^{x}$ is a function that grows rapidly as $x$ increases.

• Vertical shift: Subtracting 2 shifts the graph down by 2 units.

Step-by-Step Guidance

1. Sketch the basic graph of $y = e^{x}$.

2. Shift the entire graph downward by 2 units to obtain $y = e^{x} - 2$.

3. Label key points, such as the y-intercept and horizontal asymptote.

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Q4(b). Find $\lim\limits_{x \to -\infty} (e^{x} - 2)$.

Background

Topic: Limits at Infinity for Exponential Functions

This question tests your understanding of the behavior of exponential functions as $x$ approaches negative infinity.

Key Terms:

• Exponential decay: As $x \to -\infty$, $e^{x}$ approaches 0.

Step-by-Step Guidance

1. Recall that $\lim\limits_{x \to -\infty} e^{x} = 0$.

2. Subtract 2 from this limit to find the overall limit.

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Q5(a). Given $f(x) = x^2 + 3x - 1$, find the average rate of change of $f$ from $x = 1$ to $x = 3$.

Background

Topic: Average Rate of Change

This question tests your ability to compute the average rate of change of a function over an interval, which is similar to finding the slope of the secant line.

Key Formula:

• Average rate of change from $x = a$ to $x = b$:

$\frac{f(b) - f(a)}{b - a}$

Step-by-Step Guidance

1. Compute $f(3)$ and $f(1)$ using the given function.

2. Subtract $f(1)$ from $f(3)$.

3. Divide the result by $3 - 1$.

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Q5(b). Use limits to find the instantaneous rate of change of $f$ at $x = 1$.

Background

Topic: Instantaneous Rate of Change (Derivative Definition)

This question tests your understanding of the definition of the derivative as the instantaneous rate of change at a point.

Key Formula:

• Instantaneous rate of change at $x = a$:

$\lim\limits_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Step-by-Step Guidance

1. Write the difference quotient for $f$ at $x = 1$:

$\frac{f(1 + h) - f(1)}{h}$

2. Substitute $f(x)$ into the expression and simplify the numerator.

3. Take the limit as $h \to 0$.

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Q6. Use the definition of the derivative to find the derivative of $f(x) = x^2 - 5x$.

Background

Topic: Definition of the Derivative

This question tests your ability to use the limit definition of the derivative to find $f'(x)$.

Key Formula:

• $f'(x) = \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Step-by-Step Guidance

1. Write $f(x + h)$ and $f(x)$ using the given function.

2. Form the difference quotient $\frac{f(x + h) - f(x)}{h}$.

3. Simplify the numerator completely.

4. Take the limit as $h \to 0$.

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Q7(a). Use Basic Rules of Differentiation to find the derivative of $f(x) = x^5 - 8x^3 + 3x^2 - x + 8$.

Background

Topic: Basic Differentiation Rules

This question tests your ability to apply the power rule and sum/difference rules for derivatives.

Key Formulas:

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Sum/Difference Rule: The derivative of a sum/difference is the sum/difference of the derivatives.

Step-by-Step Guidance

1. Differentiate each term separately using the power rule.

2. Combine the results to write $f'(x)$.

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Q7(b). Find $\frac{dy}{dx}$ for $y = x + \sqrt{x} + x^{-2} - 2x^{-3}$ (Answer with radicals and fractions).

Background

Topic: Differentiation of Powers and Roots

This question tests your ability to differentiate terms with fractional and negative exponents, and to express answers in radical and fractional form.

Key Formulas:

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Derivative of $\sqrt{x}$: $\frac{1}{2\sqrt{x}}$

Step-by-Step Guidance

1. Rewrite all terms with exponents if needed (e.g., $\sqrt{x} = x^{1/2}$, $x^{-2}$, $x^{-3}$).

2. Apply the power rule to each term.

3. Express your answer using radicals and fractions as required.

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Q8(a). Find the slope of the tangent line to $f(x) = x^3 - 2$ at $x = 1$. Write the equation of the tangent line in slope-intercept form. Graph $f$ and the tangent line on the same axes. Label key information.

Background

Topic: Tangent Lines and Derivatives

This question tests your ability to find the derivative at a point (the slope of the tangent line), write the equation of the tangent line, and graph both functions.

Key Formulas:

• Slope at $x = a$: $f'(a)$

• Equation of tangent line: $y = m(x - a) + f(a)$ or $y = mx + b$

Step-by-Step Guidance

1. Find $f'(x)$ using the power rule.

2. Evaluate $f'(1)$ to get the slope $m$.

3. Find $f(1)$ to get the point of tangency.

4. Write the equation of the tangent line in slope-intercept form.

5. Sketch the graph of $f(x)$ and the tangent line, labeling the point $(1, f(1))$.

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Q8(b). Find the slope of the tangent line to $f(x) = 2x^2 + 1$ at $x = -1$. Write the equation of the tangent line in slope-intercept form. Graph $f$ and the tangent line on the same axes. Label key information.

Background

Topic: Tangent Lines and Derivatives

This question tests your ability to find the derivative at a point, write the tangent line, and graph both functions.

Key Formulas:

• Slope at $x = a$: $f'(a)$

• Equation of tangent line: $y = m(x - a) + f(a)$ or $y = mx + b$

Step-by-Step Guidance

1. Find $f'(x)$ using the power rule.

2. Evaluate $f'(-1)$ to get the slope $m$.

3. Find $f(-1)$ to get the point of tangency.

4. Write the equation of the tangent line in slope-intercept form.

5. Sketch the graph of $f(x)$ and the tangent line, labeling the point $(-1, f(-1))$.

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Q9. Find the points on the graph of $f(x) = 2x^3 - 3x^2 - 12x$ at which the tangent line is horizontal.

Background

Topic: Critical Points and Horizontal Tangents

This question tests your ability to find where the derivative is zero, indicating horizontal tangent lines.

Key Formulas:

• Horizontal tangent: $f'(x) = 0$

Step-by-Step Guidance

1. Find $f'(x)$ using the power rule.

2. Solve $f'(x) = 0$ for $x$ to find where the tangent is horizontal.

3. For each $x$ value found, compute $f(x)$ to get the corresponding point on the graph.

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Q10. The GDP of a country is $f(t) = 0.1t^3 + 0.5t^2 + 2t + 20$ (in billions), $0 \leq t \leq 4$, with $t = 0$ corresponding to 2007.

Background

Topic: Function Evaluation and Derivatives in Context

This question tests your ability to evaluate a function and its derivative in a real-world context (GDP growth).

Key Formulas:

• GDP in year $t$: $f(t)$

• Rate of change: $f'(t)$

Step-by-Step Guidance

1. For 2010, find $t$ by subtracting 2007 from 2010.

2. Evaluate $f(t)$ for that value of $t$ to find the GDP.

3. To find how fast GDP is changing, compute $f'(t)$ and evaluate at the same $t$.

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Q11. The cost to produce $x$ gaming devices is $C(x) = -0.04x^2 + 80x + 75$.

Background

Topic: Marginal Cost and Derivatives

This question tests your ability to find the derivative of a cost function and interpret the marginal cost at a specific production level.

Key Formulas:

• Marginal cost: $C'(x)$

Step-by-Step Guidance

1. Differentiate $C(x)$ to find $C'(x)$.

2. Evaluate $C'(x)$ at $x = 100$ to find the marginal cost for producing the 101st device.

3. Interpret the meaning of the marginal cost in the context of the problem.

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