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### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 7: Using the formal definition of derivative

# Limit expression for the derivative of cos(x) at a minimum point

Sal interprets a limit expression as the derivative of cos(x) at x=π, and evaluates it. Created by Sal Khan.

## Want to join the conversation?

• Ugh what is the point of learning all of this?
• Knowledge of Calculus is like a base. It's essential in every engeneering specialization, physics, astronomy etc. I'm studying logistics now, and I've never learnt Calculus or Linear Algebra (I thought that was a waste of time), now I have to learn very fast. Wish I've learnt it years ago.
So, if you have time - do Calculus.
• So, to put it straightforward: as the change of x, (Δx) approaches zero, (limit approaches zero) the secand line tends to become tangent, right?
• That is correct, the secant line between two points becomes the tangent at a single point as Δx-->0.
• After finding the slope of a tangent line, how would you find the y intercept to make a y = mx + b equation?
• You should use the point-slope form, using the point of tangency as the point. Then set x=0 and solve for y, that will be the y-intercept.
As a reminder, the point slope form is:
y - y₁ = m(x - x₁)
• I can't seem to find what to do when I got something like f(x+h) mathematically.. Really frustrates me.
• Look at the definition of f(x) and then replace every occurrence of x with whatever is in the parentheses.
Thus, if f(x) = 7x² - 4x + 11
Then, f(x+h) = 7(x+h)² - 4(x+h) + 11
And then, if appropriate, you would simplify:
In this case f(x+h) simplifies to
7x²+ 14hx + 7h² - 4x - 4h + 11
• Around he mentions we don't have the algebraic tools to figure out the function/limit algebraically. Can someone tell me where I can find out how to do this? I've always been curious how in the world you can solve problems involving trig functions that don't involve common angles without a calculator.

For example, how could I solve cos (88) without a calculator?
• A lot of times, you simply can't, there's no finite expression.

If there is, it's derived with some known starting values (such as sin 45 and sin 30 which are trivial to calculate) and then use the half-angle formulas, addition formulas and other trig identities.

This article lists values for multiples of 3 degrees:
http://en.wikipedia.org/wiki/Exact_trigonometric_constants

cos 88 = sin(90 - 88) = sin 2, and the above article mentions there's no finite algebraic expression for sin 2.
(1 vote)
• at can anyone please explain me that graph?
• The red line is the function cos(x) for all values that x can have between 0 and 2pi. At any point (x,y) on the red line the x is what you put in to the cos(x) and y is the result. So it it the graph of cos(x) = y.

Are you familiar with cosine curves? If not then you should maybe learn a bit about them before you start calculus.
• In limits, why do people keep saying "as h approaches zero" when in the end it actually will equal to zero? This confuses me very much. And how do you even get an answer when you add something to zero and subtract it by itself?
*(f(x+h)-f(x))/h*
This is how I think of it.
(a + 0) - a, which is going to equal to zero OR undefined if ((a + 0) - a)/0
I'am new to this and this might be a stupid question but any help with this will be much appreciated.
(1 vote)
• What level calculus would this be considered? (1,2, 3, or 4) or AB or BC

Thanks
• I'd say Calculus I or Introductory Calculus but I don't think it's the same everywhere.