Test 2 will cover the material in sections 2.5 through 4.3. (Note, 4.4 on the fundamental theorem of calculus will be covered on the final.) I can’t emphasize enough that this test will be difficult. There is a lot of material covered here that you have learned in this class and not previously. For a review, here are the key topics and a sample question for each. We will have 1 hour on Monday 11/30 to review for the exam.
This allows you to find $dy/dx$ when x and y are related by an equation and not a function. Here is an example.
Find $dy/dx$ and $dy^2/dx^2$ when $y^2 - x^2 = 5$.
Oh boy, word problems! Related rates boil down to the chain rule in practice. For example, if area and height are related, then the change in area and the change in height are related. Here is an example problem:
A funnel in the shape of cone has the ratio $h/r = 2$. The funnel is losing volume at the rate of 2 cm3/minute. What is the rate of change of height?
The most important thing here is the followoing fact: A continuous function on a closed interval $[a,b]$ has a maximum value and a minumum value. It achieves these at a critical point or an endpoint. As well, we learned what an absolute extrema is and how it differs from a relative extrema.
Find the extrema over $[0,3]$ of the function
$$f(x) = x \sqrt{3-x}$$
This is an important theorem for proving things in mathematics. For example, we talked about how you can prove if $f'(x) =0$ we know $f(x)$ is a constant. I won’t ask a question on this on this exam, but you may have one on the common final.
We learned what it means to call $f(x)$ increasing or decreasing on an interval $I$. We learned the first derivative test which tells us when a critical point leads to a relative extrema.
Use the first derivative test to characterize the critical points of $f(x) = (x+3)/x^2$
We learned what concavity is and what the second derivative test tells us. ($f''(x) > 0$ implies $f(x)$ is concave up).
Use the second derivative test to characterize the critical points of $f(x) = (x+3)/x^2$
Limits at infinity helped us investigate horizontal asymptotes. Remember, it took a new definition of a limit.
Find the horizontal asympotote(s) of
$$f(x) = \frac{x^2 + 2x}{\sqrt{x^4 + 2}}$$
In calculus a sketched curve should contain the following information: intercepts, asymptotes, domain, range, symmetry, relative extrema, concavity, continuity and differntiablity. Your drawing should match the mathematics. Practice with the following
Sketch
$$f(x) = | x^2 + 5x + 6| + x$$
More word problems. (Expect atleast two on the exam!) Optimization problems apply the material learned in section 3.1 to finding the largest of smallest possible values.
Let two positive numbers add to 100. What is their largest product? What is their smallest product?
(Expect a drawing on the exam. Know the area of a triangle, rectangle, square and sphere. I’ll give formulas if I use a cone, or a pyramid or some other shape.)
We learned what $dy \approx \Delta y$ and $f(x)
\approx f(c) + f'(c)(x-c)$ mean to use. Essentially, remember the tangent line approximates the function nearby. Then remember the formula for the tangent line.
Find the tangent line approximation for $f(x)$ near $x =0$ ($c$ is a constant).
$$f(x) = \frac{1}{\sqrt{1 - (x/c)^2}}$$
Use the tangent line approximation to estimate $f(.1c)$. Next, this formula comes from the theory of relativity. What is the approximate value of $f(x)$ $x$ approaches $c$?
We learned how to find answers to differential equations, what the indefinite integral and why we don’t forget to use the constant of integration.
Suppose $dv/dt = t$. Find a formula for $x(t)$. (In the notes we derived the formula when $dv/dt$ is a constant – constant acceleration. This has you find a formula when the assumption is increased acceleration.)
We learned about subscript notation, summation notation and the meaning of
$$\Sigma_1^n f(x_i) \Delta x_i$$
As well we learned the the limit of above when $n \rightarrow \infty$ is the area (if we mind the assumptions). We also defined
$$\int_a^b f(x) dx$$
to be the area under $f(x)$ between a < b provided $f(x) \geq 0$.
Find the area under $f(x) = x^3$ between $0$ and $1$. Use a Riemann sum to give the answer.