Curve Sketching
bottom: (x+3)(x+4), so VAs: x = -3, x = -4
How do we know which formula we use as our primary vs our secondary equation?
Whichever variable is being optimized (max or min)
Evaluate the following integral.
integral = 2x (6x-7) dx
4x^3 = 7x^2 + C
Find the derivative of e-x^2.
y' = -2xe-x^2
Find the derivative of (ln x)/x.
y' = (1-lnx)/x2
What do we know about the interval of a function where f'(x) > 0?
When the first derivative is positive, we know that it's increasing in this interval.
Find the differential of y = 3x^4 - 5
12x^2 dx
Solve the differential equation.
f'(x) = 9x^2- x when f(2) = 16
f(x) = 3x^2-0.5x - 6
Find the derivative of y = x2e-x.
y' = -x2e-x+2xe-x
Find the derivative of ln(2x2+1).
y' = (4x)/(2x2 +1)
What do we know about a function's interval where f''(x) < 0?
When the 2nd derivative is negative, we say that the function is concave down.
After you find both your primary and secondary equation, what is, in general, the next step?
a) draw a picture
b) find the domain limitations
c) write a new primary equation
d) find the extrema by taking the derivative and setting it equal to zero
C. you need to rewrite your primary equation using the secondary to create a single function representing both.
Use 1000 rectangles and the right-hand sum to estimate the area of the region between the graph of y = 4x - x^2 and the x-axis over the interval [1,3].
7.333333....
Write the equation of the line tangent to the graph of y = e-2x+x^2.
y - 1 = 2(x-2)
Find the derivative of ln ((1+ex)/(1-ex)).
y' = ex/(1+ex) + ex/(1-ex)
Critical numbers for either the first or second derivative are found from where the function is _________ and ________? (hint: sometimes, "never" is an answer!)
undefined and equal to zero.
The measured length of one side (s) of a wooden cube is 4 inches. The measure is correct within 0.02 inches. Estimate the propagated error in the volume of the cube. What is the relative error?
propagated error: .96 inches^3
relative error: 0.015 or 1.5%
Use 1000 rectangles and the left-hand sum to estimate the area of the region between the graph of y = x^2 + 1 and the x-axis over the interval [0,3].
11.9865
Find the integral of: e-x^4(-4x3)dx
f(x) = e-x^4 + c
Find the integral of: x2/(3-x3) dx.
f(x) = -1/3 ln|3-x3|+c
First derivative can give us three different bits of information. What are they?
intervals of increasing, intervals of decreasing, and relative extrema
What is the equation for cost?
C = Vx + F
where C = total cost, V = cost to produce one unit, x = number of units, and F = fixed costs (rent, utilities, etc).
Find the following integral: (x)/(radical 1 + 2x^2). Then, evaluate the integral from zero to two.
integral: 0.5(1 + 2x^2)^1/2
evaluates to: 1
Find the integral of: (ex^0.5)/x3.
f(x) = -0.5e-x^2 + c
Find the integral of: e2x/(1+e2x) dx.
f(x) = 1/2 ln|1+e2x|+c