Calc Exam 1

Unit Circle

Speed

distance/time

Velocity

displacement/time

Average Speed

change in distance / change in time

Average velocity

Change in position / change in time

Slope of a secant line

Instantaneous velocity

• Slope of the tangent line

• f’(x)

Instantaneous speed

• |instantaneous velocity|

• f’(x)

Limit exists if

• A limit exists from the positive direction

• Limit exists from the negative direction

• Lim f(x)- = Lim f(x)+

• (both left and right sides approach same point)

Limit does not exist if

• Both sides (+,-) lead to a different point

• Line increases/decreases without bound (infinite)

  • If f(x) increases without bound as x approaches a, the limit of f(x) as x approaches a is infinity.

  • If f(x) decreases without bound as x approaches a, the limit of f(x) as x approaches a is negative infinity.

When finding lim f(x) / g(x)

Find the pos and neg limits of the ENTIRE limits, not just each function individually

A function is continuous at c if

• f(c) is defined

• Lim F(x) exists

• Lim f(x) = f(c)

Different types of discontinuity

Removable discontinuity, jump discontinuity, vertical asymptote

Removable Discontinuity

This occurs when lim x→a of f (x) does not equal f(a).

For the function: x = a makes f(a) und, but it can be canceled out

Jump Discontinuity

Lim f(x) as x approaches a⁺ does not equal Lim f(x) as x approaches a^- (one sided lim are different)

Vertical Asymptote

At least one of lim f(x) as x approaches a⁺ or a^- = ∞ or -∞

For the function: the function is discontinious/und at a and that cant be canceled out

IVT says that

• given f is continuous on the closed interval [a,b], and N is any number between f(a) and f(b) where f(a) does not equal f(b). Then there exists at least one number c in [a,] such that f(c) = N

For IVT have to show:

1. f(a)

2. f(b)

3. f(a) < f(c) <f (b)

4. Concluding statement:

   1. IVT says that there is a number c in (a,b) such that f(c) = z

Steps to compute limit

Try to plug in x-value, if you get a real number that is the limit.

If you get 0/0 then do more work.

If you do work and still left with 0, then lim DNE

Nonzero / 0 limit

The strategy here is to look at signs of numerator and denominator

1. We know the limit dne already but we’re trying to see if it can be negative or positive infinity

2. Split up into negative and positive limit

3. See and think: as x approaches a from the neg/pos direction, are he values above the x axis?

   1. If so, pos infinity

   2. If no, neg infinity

4. Interpret:

   1. If they match, it’s that infinity (neg or pos)

   2. If not, it’s simply DNE

Ln1, ln <1, ln>1, ln<0

Ln 1 = 0

Ln < 1 = negative

Ln >1 = positive

Ln<0 = und

x²-y² =

(x-y)(x+y)

1

Squeeze Theorem

Squeeze Theorem

• If a function is trapped between two functions that both approach the same value, the middle function must also approach that value.

• g(x)f(x)h(x)

• If g(x)=h(x)=L, then f(x)=L.

When to Use It

• Try direct substitution first.

• If it doesn't work, look for a function approaching 0 × a weird/oscillating sin or cos.

• Plug the limit value into the function outside sin/cos. If it equals 0, think Squeeze Theorem.

• Remember: -1(x)1 and -1(x)1.

Steps

1. Try direct substitution.

2. Find the bounds: -1(x)1 or -1(x)1.

3. Multiply all three parts by the function outside the sin/cos.

4. Simplify to get: lower bound ≤ original function ≤ upper bound.

5. Take the limits of the lower and upper bounds.

6. If both approach the same number L, the original function also approaches L.

Example: lim x→0 of x²cos(1/x)

1. Direct substitution does not work because cos(1/x) oscillates as x approaches 0.

2. We know that -1 ≤ cos(1/x) ≤ 1.

3. Multiply everything by x²: -x² ≤ x²cos(1/x) ≤ x².

4. Find the limits of the outside functions: lim x→0 of -x² = 0 and lim x→0 of x² = 0.

5. Since both outside functions approach 0, the function in the middle is squeezed to 0.

6. Therefore, lim x→0 of x²cos(1/x) = 0.

What are limits approaching infinity representing

They represent end behavior, which also means a horizontal asymptote.

Basically, as my x value gets infinitely large/small, what value is it approaching?

Domination rules for Horizontal Asymptote

• Numerator dominates: limit = neg or pos infinity and no HA

• Neither dominates: Ratio of leading coefficient for lim and HA

• Denominator dominates: always = 0 for lim and HA

With limits to infinity, if you have an infinity/infinity function

The lim will equal ratio of leading coefficents but to prove:

• Divide by leading in term in denominator

• Or you can also simply reason it out and plug your inf/neg inf into x as seen here

A function can have at most ___ HA

2

How many VAs can a function have

infinitely many

Relationships between and f(x), f’(x), f”(x)

• If f’(x) > 0 for all x in (a,b), then f is increasing on (a,b)

• If f’(x) < 0 for all x in (a,b), then f is decreasing on (a,b)

• If f’(x) = 0 for all x in (a,b), then f is constant on (a,b)

• If f’’(x) > 0, f’(x) increasing, f(x) concave up

• If f’’(x) < 0, f’(x) increasing, f(x) concave down

Position, velocity, acceleration functions

•  Position function: gives position s(t) 

  • Units: m

• Velocity function: gives instantaneous velocity/rate of change in position

  • v(t) = s’(t) - so velocity = derivative of position function 

  • Units: m/s

• Acceleration function: gives instantaneous acceleration/rate of change of velocity

  • a(t) = v’(t) = s’’(t): so velocity = derivative of velocity function and second derivative of position function 

  • Units = m/s²

If a function is differentiable at x, is it continuous?

If a function is continuous at x, is it differentiable?

If a function is differentiable at x, it means it is also continuous at x, but not vice versa.

Types of Non-Differentiabilty

Corner, Cusp, Vertical Tangent, Discontinuity

Corner

The one sided derivatives are different

Cusp

Extreme case of a corner; slopes of the secant lines approach infinity from one side and negative infinity from the other side

Vertical Tangent

Slopes of the secant lines approach either infinity or negative infinity from both sides

Discontinuity

causes one or both of the one-sided derivatives to be nonexistent

Limit definition of derivative

Steps for tangent line

1. Find the derivative f’(x)

2. Find f’(a) = slope (m) (instantaneous rate of change)

3. f(a) = y₁

4. Plug into point slope:  y-y₁= m(x-x₁)

Derivative of a constant

0

Derivative of A linear (y = mx + b) function

m (slope)

Power Rule

(xⁿ) = nx^n-1or (axⁿ) = (an)x^n-1

Sum and difference rule

The derivative of f + or - g = the sum/diff of the derivative of f and g.

Derivative of e^x

e^x

Product Rule

 [f(x)g(x)] = f’(x)g(x) + f(x)g’(x)

Quotient Rule

(Low)(Dee High) - (high)(dee low) all dived by (low) squared

Use when functions being divided by each other