Calculus - Exam 2

What is the range of the function ​f(y)=e^y​?

(0,∞​)

Explain why the Quotient Rule is used to determine the derivative of tanx and cotx.

Both tanx and cotx are defined as ratios of two other trigonometric​ functions, tanx=sinx/cosx and cotx=cosx/sinx.

For b>​0, what are the domain and range of ​f(x)=bx​?

D = (- infinity, infinity) R = {y: y > 0}

For b>0 with b≠​1, what are the domain and range of ​f(x)=logbx and​ why?

D = (0, inf) R = (-inf, inf)

Assume the derivatives of f and g exist. How do you find the derivative of the sum of two​ functions, f+​g?

Find f′ and g′ and add them together.

If dy/dx is​ small, then small changes in x will result in relatively ____________ changes in the value of y.

If dy/dx is​ small, then small changes in x will result in relatively small changes in the value of y.

Is f(x)=x^2−3x+2x−1 differentiable at x=1​? Justify your answer.

No, the function is not differentiable because​ f(x) is not continuous at x=1.

If f is continuous at​ a, must f be differentiable at​ a?

No. If f is continuous at​ a, it is not necessarily true that the limit that defines f′ exists at a.

How do you find the fifth derivative of a​ function?

Take the derivative of the​ function, using the limit definition or other rules. The derivative of the derivative is the second derivative. Repeat this process until the fifth derivative is obtained.

Vertical Tangent Line of Circle

The Circle's outer Xs

How do you find the derivative of a constant multiplied by a function ?

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

The derivative of​ f(g(x)) equals f′(x) evaluated at​ _______ multiplied by g′(x) evaluated at​ _______.

The derivative of​ f(g(x)) equals f′(x) evaluated at g(x) multiplied by g′(x) evaluated at x.

Explain the meaning of logbx.

The expression logbx represents the power to which b must be raised to obtain x.

Why is the notation dy/dx used to represent the derivative?

The notation dy/dx is a reminder of the limit Δy/Δx (the quotient of the change in y and the change in x)

If the limit definition of a derivative can be used to find f′(x)​, then what is the purpose of using other rules to find f′(x)​?

The other rules for derivatives are easier to use and may take less time than the limit definition.

How is the property bx+y=bxby related to the property logb(xy)=logbx+logby​?

The properties are related in that each can be used to derive the other.

What is the difference between the velocity and speed of an object moving in a straight​ line?

Velocity can be positive or​ negative, depending on the direction the object is traveling. Speed is always positive.

If f is differentiable at a, must f be continuous at a?

Yes, if f is differentiable at a, then f is continuous at a.

For a given function f, what does f' represent?

f' is the slope function of f

Give an example of a function that is​ one-to-one on the entire real number line.

f(x) = 3x + 2

Slope of normal line

m = -1/f'(x)

Which of the following must be true about the value of x if x=ey​?

x>0

What is the derivative of y=e^(kx)​? For what values of k does this rule​ apply?

y′=ke^(kx) for any real number k.