This test covers lessons 14-32
In lessons 19 (trigonometric functions), 20 and 21 (chain rule), and 23 (exponential and log functions) we round out our rules for derivatives. There will be several questions on this material.
In lesson 22 (implicit derivatives) we see that we can find tangent lines to graphs given by equations, not just functions. This technique – assuming a variable depends on another in a functional manner, even if not explicitly so – is used in solving the word problems of section 3.10 related rates.
In lesson 18, a discussion of higher order derivatives is held. This will be important in lessons 31 and 32, where we learn about the relationship between concavity and the second derivative.
The derivative is interpreted in many ways:
In lesson 14 it is discussed in the context of rates of change.
In lessons 24 and 25 the rates of change are carried forward to derive equations involving rates from equations involving related variables.
in lesson 26 the tangent line view of the derivative is used to talk about approximations: $f(x+h)\approx f(x) + f'(x)\cdot h$. This is widely employed use of the derivative.
In lesson 27 and 28 the distinction between absolute maxima (over an interval) and local maxima is defined. There are two key theorems: (1) the extreme value theorem: for a continuous function $f(x)$ over $I=[a,b]$ there is an absolute maximum and absolute minimum (2) for a continuous function $f(x)$ over $I=[a,b]$ the absolute maximum (minimum) occurs at either a critical point or an endpoint.
In lesson 29 and 30 we learn – the first derivative test – to tell when a critical point will correspond to a local maximum or minimum (the derivative changes sign). This is a consequence of a simple fact: an increasing function has a postive derivative.
In lessons 31 and 32 we learn – the second derivative test – to tell a different way when a critical point will correspond to a local maximum or minimum ($f''>0$ is a local min, $f'' < 0$ a local max). This test is easier to do (sometimes) but is not always conclusive. It rests on a concept of concavity that can be defined in terms of a second derivative.
Let $f(x) = \cot(x)$. Find $f'(x)$.
Let $f(x) = e^{\sin(x)}$. Find $f'(x)$.
Let $f(x) = e^x \cdot {\sin(x)}$. Find $f'(x)$.
Let $f(x) = \log(\sin(x) + \sin(2x))$. Find $f'(x)$.
Let $f(x) = \frac{1}{\sqrt{2\pi}}e^{-(x-a)^2/2}$. Find $f'(x)$ and $f''(x)$.
Use implicit differentiation to find $dy/dx$ at $(2,3)$ to the equation $x^2 + x + y^2 = 15$. Does your answer match this plot?
Suppose $n$ and $m$ are integers. Find $dy/dx$ when $y^n = x^m$.
Find $dy/dx$ when $\sin(xy) = 1/2$.
A spy is stationed 25 feet across a street. Their eye is on a person walking along the street at 5 ft/sec. Find $d\theta/dt$ when $x=10$. (Ignore axis in graph)
As Claude walks away from a 16 foot lampost, the tip of his shadow moves twice as fast as he does. What is Claude's height?
If $f(x) = e^x + e{-2x}$, find $f''(3)$.
Use an approximation to find $\sqrt{17}$.
Estimate the change in $y$ for a $1$ unit change in $x$ for $f(x) = x^3 - 2x$ at $x=3$.
Let $f(x) = x^3 − 3x + 1$. Find the critical points of $f$.
Let $f(x) = x\cdot e^{-x}$. Find the critical points of $.
Let $f(x) = \sin(x) + \cos(x)$. Find the critical points of $f$ in $[0, 2\pi]$.
Using calculus, find the maximum and minimum value of $f(x) = \tan(x) - 2x$ on $[0,1]$.
Using calculus, find the maximum and minimum value of $f(x) = x - 4x/(x+1)$ over $[0,3]$.
Using calculus, find the minimum value of $f(x) = 10^{-16}/x + x$ for $x > 0$. (It happens at a critical point.)
A sign chart is given for $f'(x)$:
+ 0 + 0 - 0 + f'(x) ------ 2 ------ 4 -------- 6 ------
What are the critical points of $f(x)$? Which are local maxima? Which are local minima? Which are neither?
From the graph estimate values of the critical points of $f(x)$
From the graph, on what intervals is $f'(x) > 0$?
Below is the graph of $f'(x)$ for some function $f(x)$. Based on the graph, what are the critical points of $f(x)$?
From this graph, characterize the critical points as either: local maxima, local minima, or neither using the first derivative test.
Consider the two graphs of funcitons below, is it possible that the second one is the first derivative of the first? Explain what you compare to see.
The following graph is of $f'(x)$ – and NOT $f(x)$.
What are the critical points of $f(x)$? Which critical points are relative maxima of $f(x)$? On what intervals do you know $f(x)$ is increasing? On what intervals if $f(x)$ concave up?
Consider the graph of $f(x) = x^x$ over $[0,2]$ ($f(0)=1)$. Estimate the value $c$ guaranteed by the mean value theorem from the graph:
The Mean Value Theorem says for nice function $f(x+h) - f(x) \approx f'(x) \cdot h$.
The extreme value theorem says for $f(x)$ continous on $[a,b]$ there exists a $c$ in $[a,b]$ where $c$ is a critical point
For a continuous function on $[a,b]$, we know that the absolute maximum occurs at and endpoint or at a critical point
Rolle's theorem says that if $f$ is continous on $[a,b]$ and differentiable on $(a,b)$ and $f(a) = f(b)$, then there is some $c$ where $f'(c) = 0$.
Rolle's theorem is a special case of the Mean Value Theorem, as the slope of the secant line is just $0$ under the assumptions.
A function is increasing on $(a,b)$ if $f'(x) > 0$ on $(a,b)$.
A differentiable function has a positive derivative on $(a,b)$ if it is *increasing:
A function has a derivative that changes sign from $-$ to $+$ at $x=c$. Is $c$, a critical point, a relative maximum?
At $x=c$, $f(x)$ has a critical point. It is found that $f''(c)=0$. Is $c$ possibly a relative maximum?