How do you find the limit of a composite function when there is discontinuity? $$\lim_{x\rightarrow c}f\left(g\left(x\right)\right)=$$

1)Find the $$\lim_{x\rightarrow c}g\left(x\right)=d$$ . Determine if the limit is approaching the y-value, d, from vertically above or vertically below. 2)Evaluate the $$\lim_{x\rightarrow d^+}f\left(x\right)$$ (if approached from vertically above) or $$x\rightarrow d^-$$ (if approached from vertically below).

What is the Intermediate Value Theorem?

A function that is continuous on a closed interval [a, b] takes on every value between f(a) and f(b). In other words, if f(w) is between f(a) and f(b), then w is within [a, b].

What is the average rate of change?

* Slope of the secant line * Approximation of the derivative

What is the formula for the average rate of change?

* $$\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

What is the instantaneous rate of change?

* Slope of the tangent line * Slope of the function * Derivative

What is the formal definition of the derivative?

$$\lim_{h\rightarrow0}\ \frac{f\left(x+h\right)-f\left(x\right)}{h}$$

What is the alternative definition of the derivative?

$$\lim_{x\rightarrow c}\ \frac{f\left(x\right)-f\left(c\right)}{x-c}$$ Great for showing if a piecewise function is differential at the connection point. Use 1-sided limit with correct piece.

$$\frac{d}{dx}\left(\sec\ u\right)=$$

$$\sec\ u\ \tan u\ \frac{du}{dx}$$

$$\frac{d}{dx}\left(\csc\ u\right)=$$

$$-\csc\ u\ \cot u\ \frac{du}{dx}$$

$$\frac{d}{dx}\left(a^u\right)=$$

$$a^u\left(\ln a\right)\ \frac{du}{dx}$$ (copy paste)(natural log of the base) (derivative of exponent)

$$\frac{d}{dx}\arcsin\ u=$$

$$\frac{1}{\sqrt[]{1-u^2}}\ \frac{du}{dx}$$

$$\frac{d}{dx}\arccos\ u=$$

$$\frac{-1}{\sqrt[]{1-u^2}}\ \frac{du}{dx}$$

$$\frac{d}{dx}\operatorname{arcsec}\ u=$$

$$\frac{1}{\left|u\right|\sqrt[]{u^2-1}}\ \frac{du}{dx}$$

$$\frac{d}{dx}\operatorname{arccsc}\ u=$$

$$\frac{-1}{\left|u\right|\sqrt[]{u^2-1}}\ \frac{du}{dx}$$

$$\frac{d}{dx}\arctan\ u=$$

$$\frac{1}{1+u^2}\ \frac{du}{dx}$$

$$\frac{d}{dx}\operatorname{arccot}\ u=$$

$$\frac{-1}{1+u^2}\ \frac{du}{dx}$$

$$\frac{d}{dx}\left(\log_bu\right)=$$

$$\frac{1}{u\ \ln\ b}\ \frac{du}{dx}$$

How do you do Implicit Differentiation?

When you can’t isolate y in terms of x, it’s time to take the derivative implicitly. 1) Differentiate both sides of the equation with respect to x. Any time you take a derivative of y label it dy/dx . (This is the Chain Rule.) 2) Collect the terms with dy/dx on one side of the equation. 3) Factor out dy/dx. 4) Solve for dy/dx by dividing.

How do you do logarithmic differentiation?

1) Take the natural log of each side. 2) Use properties of logarithms to rewrite each side. 3) Take the derivative of each side. (Usually need to use the Product Rule on one side.) 4) Factor out dy/dx. 5) Solve for dy/dx by dividing.

If f(x) and g(x) are inverses of each other, how do you find g'(a)?

$$g$$ Must remember that the x-value for g is NOT the x-value for f. So, (a, b) for g becomes (b, a) for f. Use the x =b with f.

How do you find velocity?

v(t) = s'(t) First derivative Units: ft/sec

How do you find acceleration?

a(t) = v '(t) = s ''(t) Second derivative of position or First derivative of velocity Units: $$\frac{ft}{\sec^2}$$

How do you know when an object is moving forward, right or upward.

$$s$$ $$v\left(t\right)>0$$ Moving to the right or forwards is the same thing as determining when the function is increasing.

How do you know when an object is moving backwards, left or downward?

$$s$$ $$v\left(t\right) Moving to the backwards, left or downward is the same thing as determining when the function is decreasing.

When is an object is changing directions?

Sign of v ( t ) changes (crosses the t-axis)

When is an object at a maximum height or minimum height?

$$s$$ Changing from + to - or - to +

When does an object have a maximum velocity or minimum velocity?

$$v$$ Changing from + to - or - to +

When does an object have a maximum acceleration or minimum acceleration?

$$a$$ Changing from + to - or - to +

What i s Displacement?

$$∆y=s(b)-s(a)$$ Change in the y-values

How do you find Average Velocity?

$$\frac{s(b)-s(a)\ }{b-a}$$ Average velocity is the slope of the secant line.

Speed=|Velocity| Maximum speed is the farthest point on the velocity graph from the t-axis.

When is an object is slowing down?

Speed is decreasing (slowing down) when the object is moving towards the t-axis. Velocity and acceleration have different signs

When is an object is s speeding up?

Speed is increasing (speeding up) when the object is moving away from the t-axis. Velocity and acceleration have the same signs

What is linearization?

Linear approximation – Using the equation of a tangent to approximate a curve. Need: Point & Slope (derivative) Use: $$y-y_1=m(x-x_1)$$ Rewrite: $$f(a)\approx y_1+m(a-x_1)$$

What is L'Hopital's Rule

$$\lim_{x\rightarrow c}\ \frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\rightarrow c}\ \frac{f$$ Use when $$\lim_{x\rightarrow c}\ \frac{f\left(x\right)}{g\left(x\right)}=\ \frac{0}{0}\ \ \ OR\ \ \ \frac{\infty}{\infty}$$ Remember to show the limit of the f(x) equals 0 and the limit of g(x) equals 0.

What is Rolle's Theorem?

If a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b) and has the same y-value at the endpoints a and b, then there must be at least one value of x, call it c, between a and b where the function has a horizontal tangent (max or min point).

AB Sem 1 Flashcards

Student preview

64 questions

Show all answers

1.

FLASHCARD QUESTION

Front

How do you find the limit?

Back

It is based upon degree.

If the DoN < DoD, then the limit is approaching 0 (because the Horiz Asymptote is y = 0)

If the DoN = DoD, then the limit is approaching the LC/LC (because the Horiz Asymptote is y = LC/LC)

If the DoN > DoD, then you must think about the End Behavior Model (Leading term)/(Leading term)

What is the limit of the End Behavior Model? That is the answer. (Either infinity or negative infinity because it is a slant asymptote.)

2.

FLASHCARD QUESTION

Front

If you find a limit and get a #/0, what do you know?

Back

There exists a Vertical asymptote. So, you know that the limit is approaching infinity, negative infinity or DNE. If the limit is approaching c from 1-side, take a value close to c and substitute into the function. If you get a positive value, the limit is approaching a positive infinity. If negative value, the limit is approaching a negative infinity.

3.

FLASHCARD QUESTION

Front

The limit of a constant is a ...

Back

constant.

4.

FLASHCARD QUESTION

Front

The derivative of a constant is ...

Back

zero.

5.

FLASHCARD QUESTION

Front

How do you find the limit of a piecewise function?

Back

Find the 1-sided limit of each piece where the pieces meet at x = c.

6.

FLASHCARD QUESTION

Front

What is the continuity test?

Back

7.

FLASHCARD QUESTION

Front

What is the Squeeze Theorem?

Back

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