Another way to do this problem without calculus. Find the equation of the line that passes through (-3,3) and is perpendicular to the given line (the slope of the perpendicular must be 3/5. Once you have the two equations find the point of intersection (perhaps by the substitution method). https://www...

A good strategy for solving this problem might be to make up a specific example and think about how you would go about it then (for example let a=36). That said, we can write the equation of the tangent line passing through (x0, y0) by finding y' (implicitly), getting the slope, and then using point-slope...

A good strategy for solving this problem might be to make up a specific example and think about how you would go about it then (for example let a=36).
That said, we can write the equation of the tangent line passing through (x0, y0) by finding y' (implicitly), getting the slope,...

When x= 0, y = +/- 1/2. So find the equation of the tangent line at (0,1/2) and (0, -1/2).
Finding an explicit formula for y' using implicit differentiation is a formidable task. Instead just use implicit differentiation, leaving the result as you find it. Then substitute 0 for...

Hi Abigail.
How about I give you some hints to get you started. You will need formulas for the volume of a right circular cylinder and a sphere. You will also need a formula for the surface area of those objects. The problem does not state whether the "floor" of the observatory...

Hi Rahul,
This problem is complicated. There is a cool trick required. Think of the equation of the tangent line as y = mx +k (The suggestion is to use k as the value for the y-intercept because you will be making use of the formula for the discriminant of a quadratic formula...

Hi Daniel, My guess is that you perhaps have not learned how to find the derivative of a function. The intent of the problem likely is to evaluate the limit directly. If that is the case:
The idea is to find something we can multiply numerator and denominator by so that the "h"...

Hi April,
The given info is the same as telling you the function (line) passes through the points (0,7) and (3,1). Calculate the slope between those two points, then use either point-slope or slope-intercept to write the equation of the line.
Once you have...

Since the directions indicate you must use theta between -pi/2 and pi/2 when (x,y) is in the 2nd or 3rd quadrant you will have to use the negative square root of X2 + y2.
For part a) r = sqrt(4+9) = sqrt(13).
Theta = tan-1(y/x) = tan-1(-3/2) = -0.983 radians...

You should have a theorem in your text book indicating that for x n the radius of convergence you can find MacLaurin for f(t) in this case t2et^2, then integrate that series term by to get a power series representation for the integral in question. The assumption here is that you know how to find...

Hi Andrew,
Here are some hints (assuming you have not already figured it out). Use some coordinate geometry, assigning coordinates to the vertices of the triangle, write some equations for the sides of the outside triangle and this "inside" triangle. The difficult part is...

f(0)=1, so a+bcos0 = 1, or A) a+b = 1.
f(pi/4) = 3/2, so B) a+bcos(cpi/4) = 3/2
Let g(x) equal the derivative of f. g(x) = -bcsincx.
g(pi/4) = 3/2, so C) -bcsin(cpi/4) = 3/2.
Getting ugly. At this point there are 3 equations...

Hi Emily,
Not sure which "formulas" you have learned at this point. Parallel lines have the same slope.
The slope of the given line is found by solving for y (slope-intercept form).
So, y = -4x + 5 tells you the slope of the line is...

Hi Imani,
Do you know the three identities for the cosine of twice an angle? If not, take time to locate them in your trig book.
One of them is cos2x = 1-2sin2x. Since you are given sinx = 9/41, substitute that value into the aforementioned identity and simplify.

Hi Julia,
Are you using formula for area of equilateral triangle is A = s2√3/4?
If so, then dA/dt = 2s√3/4 ds/dt= s√3/2 ds/dt.
Now try substituting given values into that equation.

Likely the expectation is for you to use the identity for the tangent of the sum of two angles.
tan(x+y) = (tan x + tan y)/(1-tanxtany).
In this case x = arc sin(3/7) and y = arc cos 4/5.
To find the tangent of x and tangent of y there are...

Making a table of values can be a pain. Calculator optional, but highly desirable. Check out the following graph to get the idea.
https://www.desmos.com/calculator/otd8ormj0f
Enable and disable the functions for f, g, h to get an idea of what the point is.

Hi Notroe,
Whenever I see a problem like this (asked to solve for x and four terms on the left side and 0 on the right) I get suspicious that factoring by grouping might lead to something.
Try:
2sinx(2cosx - √3) - √2(2cosx - √3) = 0 (notice...

Hi Julia,
Graphing the function helps to visualize how to apply the Intermediate Value Theorem to locate the fact that there are two real roots between 0 and 2 (you just cannot use the endpoints of that interval to prove the point).
Take a look at: https://www...

Hi Don,
If this is written as f(x) = 2e-5x^2, then you can ask the question does f(x) = f(-x). If so this is an even function and is symmetric about the y-axis.
Find f'(x) to determine increasing/decreasing. f'(x) = -20xe-5x^2. This will only equal 0 when x =...