Calculus 1 (set 1)
False, the function may be discontinuous or undefined at x
(T or F) The limit as f approaches x is always f(x)
cup
A concave up curve is shaped like a
denominator is != 0
A rational function is continuous everywhere that its:
Break the function into pieces and find the limit of the pieces
An idea behind finding the limit of a function is to
f''(x) switches signs
An inflection point is where
End points, stationary points, singular points
Critical points may be
instantaneous rates of change
Differential calculus is all about
No
Does f(x) have to be defined at c in order for the limit to exist?
-sin x
Dx cos(x) =
cos x
Dx sin(x) =
Between two points on a function, somewhere there is a point where the slope of the secant lie between the two points is equal to the slope of the tangent line
Explain the mean value theorem
differentiation
Finding a derivative is called:
1.) Solve for f''(x) 2.) Solve for f''(x) > 0 and f(''(x) < 0
How do you determine where a function is concave up or concave down
1.) Find the antiderivative of the f(x) 2.) Evaluate F(upper limit) - F(lower limit)
How do you evaluate a simple definite integral?
1.) Instead of putting the x value into the derivative formula, put in a variable such as c 2.) Solve for the functions 3.) Factor out the denominator
How do you find an equation that will give you f'(x) for any x value rather than solving for a specific point?
1.) Find f' 2.) Set f' = 0 and solve
How do you find stationary points?
1/2 * radius² * angle of sector
How do you find the area of a sector of a circle?
1) Find the first derivative of f(x). 2) Plug x value of the indicated point into f '(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line.
How do you find the equation of a tangent line at a given point on a graph?
1) Find the first derivative of f(x). 2) Plug x value of the indicated point into f '(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line.
How do you find the equation of the tangent line using the derivative
1.) Find the critical points 2.) Evaluate f at the critical points, highest is max, lowest is min
How do you find the local maximum and minimum?
Find the formula for the instantaneous rate of change, set this formula equal to the desired y output and solve for the variable
How do you find the the x input that corresponds to a given y output?
Vertical: Where denominator = 0 Horizontal: the Limit as x→∞
How do you find the vertical and horizontal asymptotes of a function?
1.) take the derivative of f(x) 2.) Find the zeros of f'(x) 3.) Use test points to test in between the intervals to see where f(x) is positive or negative / set f'(x) > 0 and f'(x) < 0 and solve for the inequalities
How do you find where x is increasing or decreasing on an interval
1.) Take derivative of both sides 2.) Using the *product rule*, separate into two functions (3x and y) 3.) Find the derivative of each function and plug into the product rule
How do you solve for Dx 3xy = 2
1.) Seperate x and y terms 2.) Integrate both sides to find function (Don't forget constant) 3.) Plug in point to solve for constant
How do you solve for a differential equation using separation of variables?
Using the chain rule where 2x is the inner and sin(x) is the outer
How do you solve for: dx sin(2x)
1.) Take the derivative of the function 2.) Find Δy/Δx (secant line between the interval points) 3.) Put secant line slope equal to the derivative 4.) Solve for variables
How do you use the mean value theorem?
1.) Make a guess for x by plugging it in to f(x) and seeing if = 0 2.) Find the tangent line using the derivative 3.) Find where the tangent line intersects the x axis (where y = 0) 4.) Repeat using the x axis / tangent line intersection as the new "guess" input until sufficiently close to 0
How does Newtons method work?
The "exponent" on the value being approached (c) is a - if approaching from values less than c, and a + if approaching from values greater than c
How is the direction that a limit is being approached from denoted?
1.) Factor numerator 2.) Cancel (x-4) from numerator and denominator 3.) Input 4 into new function 4.) f(4) = 8
How should f be defined at x=4 in order to make it continuous there?
The limit is 3 because the function is discontinuous
If F(-.01) = 2.99 and f(.01) = 2.99 but f(0) = -6 What is the limit and why?
Does not exist (∞)
If a function has a vertical asymptote, the limit as x approaches that asymptote is:
Because if the denominator in Δy/Δx is 0, there will be no answer
If as the change in x approaches 0, you get closer to the derivative, then why can you not just input a 0 change in x and calculate?
factor/simplify the function algebraically, then input c
If f(x) is undefined at c, how can you find the limit *without creating a table*?
By creating a table and inputting values close to c, by graphing f(x), or by using algebra to finish the
If f(x) is undefined at c, how can you find the limit?
Local extreme
If the derivative of a function is positive on one side of a critical point, and negative on the other, the critical point is a
Does not exist
If the limit of a function as it goes to 1 is ∞ from the right side, but is -∞ from the right side, what is the limit?
The limit does not exist
If the limit of a function as x approaches c from the left is 3, but when it approaches from the right it is 5, what is the limit?
application of the chain rule
Implicit derivation is simply an
No
Is a critical point always a local extreme
Horizontal asymptotes
Limits as x approaches infinity are:
an element (a little bit) of x
Put simply, dx means
All real numbers
Sine and cosine are continuous at:
check your shit
Slow down and
false
T or F a derivative is the same as a differential
False, must use the quotient rule
T or F f(x)/g(x) = f'(x)/g'(x)
0
The derivative of a constant function is
another function that describes the rate at which a dependent variable changes with respect to the rate at which an independent variable changes
The derivative of a function is
derivative
The instantaneous slope is the
the first step in differentiation
The last step in calculation corresponds to
the limit raised to the power of n
The limit of a function raised to the power of n is equal to
Derivative
The limit of the secant line as the difference between the points (h) approaches 0 is the:
Δy/Δx = y's per x
The slope of a line is equal to
= x - f(x)/f'(x)
The x intercept of the tangent line (for Newton's method) is equal to
find the derivative and multiply it by dx
To find the differential you
divide the numerator and denominator by x to the highest power of x that appears in the denominator
To find the limit as x approaches infinity of a rational function you may:
1.) Differentiate both sides, differentiating x terms as normal 2.) Differentiate the *Y ONLY* terms and add y' next to each 3.) Differentiate x and y terms using product/quotient rule, add y' next to each y term differentiated 4.) Isolate y' terms 5.) Solve for y'
What are the steps for implicit differentiation?
Finding the roots (zeros) of a function
What does Newtons method for?
The sum of all the elements (little bits) of x
What does the integral mean?
Continuous on the closed interval [-7, 7]
What interval is this function continuous on? Is it open or closed?
Any equation in which the unknown is a function
What is a differential equation?
A discontinuity that cannot be defined by any method
What is a non-removable discontinuity?
A discontinuity that can be defined by finding the limit through factoring, theorems, etc...
What is a removable discontinuity?
A straight line between two points on a curved one f(x + h) - f(x) / h
What is a secant line?
A point where the graph of f has a sharp corner, vertical tangent, or jump
What is a singular point?
if f'(c) = 0, c is stationary point (tangent line is horizontal)
What is a stationary point?
A point where f is concave up on one side of c, and concave down on the other side
What is an inflection point
It is the slope of a tangent line to a curve, or, an instantaneous rate of change
What is an interpretation of the derivative?
y - y1 = slope(x - x1) P(x1, y1)
What is point slope form?
4 1.) multiply the entire fraction by 4, and put a 4 in the denominator (since only the numerator will be multiplied by 4 this is equivalent) 2.) Recall that lim c*f(x) = c * lim f(x) and move the 4 in front of the limit 3.) Factor a 4 from the numerator and denominator, resulting in the limit of the special trig function equal to 1 4.) 4 * 1 = 4
What is the answer, how do you solve it?
The derivative of a composite function is the *derivative of the outside function evaluated at the inner function, times the derivative of the inner function* (f o g)' (x) = f'(g(x))*g'(x)
What is the chain rule?
A constant multiplier k can be passed across the operator Dx Dx[k * f(x)] = k* Dx (f(x)) The derivative of a constant multiplied by a function is the same as the the constant multiplied by the derivative
What is the constant multiplier rule?
As x in the function f(x) approaches some constant c, f(x) approaches the limit or A limit is the value that a function or sequence "approaches" as the input or index approaches some value.
What is the definition of a limit?
lim x→c f(x) - f(c) / x - c
What is the derivative formula where the limit of one x input it getting closer to another x input is being taken?
7
What is the derivative of 7x ?
1.) Take derivative 2.) Use derivative to find critical points 3.) Determine where f is increasing/decreasing 4.) If f'(x) increasing on one side of the point, and decreasing on the other, the point is a local extreme 5.) Plug points into f(x) to determine extremes
What is the first derivative test?
Take the derivative of both sides
What is the first step of implicit differentiation?
f(x + Δx) = f(x) + f'(x)(Δx)
What is the general formula for linear approximation
∫(f(x))dx = (f(x)^r+1)/r+1
What is the generalized power rule?
there exists some c in the interval where Δy/Δx = f'(x)
What is the mean value theorem
dx/dy
What is the notation for derivative?
If f(x) = x^n then f'(x) = nx^(n-1) the derivative of x to the power of in, is n times x to the power of n minus 1
What is the power rule?
Dx[f(x)g(x] = g(x)Dxf(x) + f(x)Dxg(x) The derivative of the product of two functions is equal to function f times the derivative of function g, plus vice versa
What is the product rule?
To find the original function from the derivative
What is the purpose of the antiderivative?
(f / g)' (x) = (g(x)f'(x) - f(x)g'(x)) / g^2(x) denominator function stays denominator
What is the quotient rule?
1.) Find the *STATIONARY* points using the derivative 2.) Take the second derivative 3.) If f''(c) < 0 then f(c) is a local maximum of f, If f''(c) > 0 then f(c) is a local minimum of f *note that >0 is min and <0 is max*
What is the second derivative test?
The limit of f(x) as x->c is equal to f(c) provided that f(c) is defined.
What is the substitution theorem
where f''(x) = 0 or is undefined
What points are candidates for points of inflection?
When f''(c) = 0 at a stationary point
When does the second derivative test fail?
Whenever you differentiate a y term, since implicit differentiation requires you to differentiate all terms/sides, any y term will require the chain rule but some terms will require product rule, etc... before hand
When performing an implicit differentiation when do you apply the chain rule?
the greatest *integer* less than or equal to x
[[x]] denotes
the derivative of whatever follows with respect to x
d/dx means
1.) Put into exponent form: x^1/2 2.) Apply power rule: 1/2 * x^-1/2 3.) Put back into root form: 1/√x 4.) Multiply fractions: 1/2 * 1/√x = 1/2√x 1/2√x
d/dx √x
How do you find the limit as x→∞ of a rational function
divide the numerator and denominator by x to the highest power in the denominator
-13x^-14
dx/dy f(x) = x^-13
differential
dy means
The ratio of the change in y per x
dy/dx means
you can sketch its graph without lifting your pencil
f(x) is continuous if:
(√x + 1)(√x - 1)
factor (x - 1)
lim f(x) + lim g(x)
lim [ f(x) + g(x) ] =
lim f(x) * lim g(x)
lim f(x) * g(x) =
lim [ f(x) - g(x) ]
lim f(x) - lim g(x) =
lim f(x) / lim g(x)
lim f(x)/g(x) =
0
lim x→0 1-cos(t)/t =
1
lim x→0 sin(t)/t =
√lim f(x)
lim √f(x) =
