Step-by-Step Calculus Study Guide for MAT 131 Exam 2

Q1. Find the first derivative of the following functions. Do not leave answers with negative exponents or fractions which contain fractions. Simplify your final answer.

Background

Topic: Differentiation

This question tests your ability to compute the derivative of various functions using rules such as the power rule, product rule, quotient rule, and chain rule. Simplification is also required.

Key Terms and Formulas

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

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Step-by-Step Guidance

1. Identify the type of function (polynomial, trigonometric, exponential, logarithmic, etc.) and which differentiation rule(s) apply.

2. Apply the appropriate rule(s) to each term. For example, use the power rule for $x^n$, the product rule for products, and the quotient rule for quotients.

3. For composite functions, use the chain rule to differentiate the outer function and multiply by the derivative of the inner function.

4. After differentiating, simplify your result. Make sure to rewrite any negative exponents as positive and simplify complex fractions.

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Q6. If $f$ is a one-to-one function with $f(3) = 5$ and $f'(3) = 10$, find the derivative of $f^{-1}$ at $x = 5$.

Background

Topic: Derivatives of Inverse Functions

This question tests your understanding of how to find the derivative of the inverse of a function at a specific point, given information about the original function and its derivative.

Key Terms and Formulas

• Inverse Function Derivative Formula: $\left(f^{-1}\right)'(a) = \frac{1}{f'(f^{-1}(a))}$

Step-by-Step Guidance

1. Recall that if $f$ is one-to-one and differentiable, then the derivative of its inverse at $a$ is $\left(f^{-1}\right)'(a) = \frac{1}{f'(f^{-1}(a))}$.

2. Identify $a = 5$ and use the given $f(3) = 5$ to find $f^{-1}(5) = 3$.

3. Substitute $f^{-1}(5) = 3$ into the formula: $\left(f^{-1}\right)'(5) = \frac{1}{f'(3)}$.

4. Use the given $f'(3) = 10$ to set up the final expression for the derivative.

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Q8(a). The volume of a sphere is $V = \frac{4}{3}\pi r^3$. If the rate of change of the volume is 9 cubic inches per second, find the rate of change of the radius when $r = 3$ inches.

Background

Topic: Related Rates

This question tests your ability to relate the rates of change of different quantities using implicit differentiation with respect to time.

Key Terms and Formulas

• Volume of a Sphere: $V = \frac{4}{3}\pi r^3$

• Related Rates: $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related via differentiation.

Step-by-Step Guidance

1. Differentiate both sides of $V = \frac{4}{3}\pi r^3$ with respect to $t$ to relate $\frac{dV}{dt}$ and $\frac{dr}{dt}$.

2. Apply the chain rule: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

3. Plug in the given values: $\frac{dV}{dt} = 9$ and $r = 3$.

4. Set up the equation to solve for $\frac{dr}{dt}$, but do not compute the final value yet.

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Q9. Suppose a particle is moving according to the position function $s(t) = 5t^2 - 10t + 6$, where $s$ is measured in feet. Find the velocity and acceleration functions.

Background

Topic: Motion Along a Line (Kinematics)

This question tests your ability to find velocity and acceleration by differentiating the position function with respect to time.

Key Terms and Formulas

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

Step-by-Step Guidance

1. Differentiate $s(t)$ with respect to $t$ to find the velocity function $v(t)$.

2. Differentiate $v(t)$ with respect to $t$ to find the acceleration function $a(t)$.

3. Write both functions in simplified form.

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Q10. Suppose the cost of producing $x$ units of a product, in dollars, is given by the function $C(x) = \frac{1}{4}x^2 - 20x + 150$. Find the average cost and the marginal cost of producing the first 10 units.

Background

Topic: Marginal Analysis in Economics

This question tests your understanding of average cost and marginal cost, which are important concepts in economics and business calculus.

Key Terms and Formulas

• Average Cost: $AC(x) = \frac{C(x)}{x}$

• Marginal Cost: $MC(x) = C'(x)$

Step-by-Step Guidance

1. Compute the average cost for $x = 10$ by evaluating $AC(10) = \frac{C(10)}{10}$.

2. Find the derivative $C'(x)$ to get the marginal cost function.

3. Evaluate $C'(10)$ to find the marginal cost at $x = 10$.

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