Calc 1 Final Exam Review
Welcome to the Calc 1 Final Exam Review! This page will be your hub for all the resources you need to prepare for the final. We'll be covering topics from limits to integration, with a focus on problem-solving and key concepts. Let's get started!
Limits and Continuity
Let's start with a graphical approach to limits. Consider the function $f(x)$ whose graph is provided. We want to determine the following limits:
- $lim_{x \to \infty} f(x)$
- $lim_{x \to -\infty} f(x)$
- $lim_{x \to 1} f(x)$
- $lim_{x \to 3} f(x)$
Also, identify any asymptotes of the function. Remember that the limit exists if the left-hand limit and the right-hand limit are equal. Asymptotes occur where the function approaches infinity or negative infinity.
Differentiation
Differentiation is a fundamental concept in calculus. Here are some examples to practice your differentiation skills:
- Find the derivative of $g(x) = \frac{5}{2}x^2 - 8x + 17$. Remember the power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$.
- Differentiate $f(x) = (x + 7\sqrt{x})e^x$. You'll need the product rule: $\frac{d}{dx}(uv) = u'v + uv'$. Also, recall that $\sqrt{x} = x^{1/2}$.
- Find $m'(x)$ if $m(x) = \frac{\sqrt{x}}{x-3}$. Apply the quotient rule: $\frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2}$.
- Differentiate $y = \frac{2x}{9 - tan(x)}$. Remember the derivative of $tan(x)$ is $sec^2(x)$.
- Find the derivative of $n(x) = (\frac{x-2}{x+2})^8$. Use the chain rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$.
Also, consider implicit differentiation: Find the equation of the tangent line to the curve $x^2 + 6xy + 12y^2 = 28$ at the point (2, 1). Remember to differentiate both sides with respect to $x$ and solve for $\frac{dy}{dx}$.
Limits and L'Hôpital's Rule
Evaluate the limit: $lim_{x \to 1} \frac{sin(x - 1)}{x^3 + 2x - 3}$. If direct substitution yields an indeterminate form (0/0), consider using L'Hôpital's Rule: $lim_{x \to a} \frac{f(x)}{g(x)} = lim_{x \to a} \frac{f'(x)}{g'(x)}$.
Curve Sketching
Given the function $f(x) = \frac{x-4}{x^2}$, answer the following questions to sketch the curve:
- What is the domain of the function?
- What are the x-intercepts?
- What are the y-intercepts?
- Find the interval(s) on which $f$ is increasing.
- Find the interval(s) on which $f$ is decreasing.
- Find the interval(s) on which $f$ is concave up.
- Find the interval(s) on which $f$ is concave down.
- Find the local maximum for $f$.
- Find the local minimum for $f$.
Optimization
A classic optimization problem: A box with an open top is to be constructed from a 7 ft by 6 ft rectangular piece of cardboard by cutting out squares from the corners and bending up the sides. Find the largest volume the box can have. Set up a volume function and find its maximum using calculus.
Related Rates
A boat leaves the dock at 1pm and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 2pm. How many minutes after 1pm were the two boats closest together? Use the Pythagorean theorem to relate the distances and differentiate with respect to time.
Integration
Find the following integrals:
- $\int_1^{16} x^{-3/4} dx$
- $\int (x^{1.3} + 11x^{4.5}) dx$
- $\int \frac{e^x}{(7 - e^x)^2} dx$
- $\int x^3 cos(x^4+2) dx$
- $\int \frac{x^3}{x^4 - 5} dx$
- $\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx$
- $\int_0^1 (1-x)^9 dx$
- $\int_4^4 (2x - \sqrt{16-x^2}) dx$
- $\int_0^1 sin(3\pi t) dt$
Good luck with your final exam!