Differential Calculus Exam Questions with Verified Solutions (Graded A+)
Average Rate of Change - Answers Slope of two points
Instantaneous Rate of Change - Answers Derivative
Derivative is - Answers the slope of the tangent line at one point
Mean Value Theorem - Answers If a function is both continuous and differentiable on an interval, then
there exists some 'c' in that interval at which the average rate of change equals the instantaneous rate
of change
Normal line - Answers Perpendicular to a tangent line
Definition of Derivative - Answers The limit as h approaches 0 of [f(x+h) - f(x)]/h
A function is not differentiable at - Answers A corner, cusp, vertical tangent, or discontinuity (point,
jump, or infinite discontinuity)
Product rule - Answers "First times the derivative of the second plus second times the derivative of the
first"
Quotient rule - Answers "Bottom times the derivative of the top minus top times the derivative of the
bottom, all over the bottom squared"
Chain rule - Answers "Outside-Inside" Rule
d/dx sin(x) - Answers cos(x)
d/dx cos(x) - Answers -sin(x)
d/dx tan(x) - Answers sec²(x)
d/dx sec(x) - Answers sec(x)tan(x)
d/dx csc(x) - Answers -csc(x)cot(x)
d/dx cot(x) - Answers -csc²(x)
d/dx arctan(x) - Answers 1 / (1+x²)
d/dx ln(x) - Answers 1 / x
d/dx e^x - Answers e^x
Velocity - Answers Derivative of position
Acceleration - Answers Derivative of velocity
Average Rate of Change - Answers Slope of two points
Instantaneous Rate of Change - Answers Derivative
Derivative is - Answers the slope of the tangent line at one point
Mean Value Theorem - Answers If a function is both continuous and differentiable on an interval, then
there exists some 'c' in that interval at which the average rate of change equals the instantaneous rate
of change
Normal line - Answers Perpendicular to a tangent line
Definition of Derivative - Answers The limit as h approaches 0 of [f(x+h) - f(x)]/h
A function is not differentiable at - Answers A corner, cusp, vertical tangent, or discontinuity (point,
jump, or infinite discontinuity)
Product rule - Answers "First times the derivative of the second plus second times the derivative of the
first"
Quotient rule - Answers "Bottom times the derivative of the top minus top times the derivative of the
bottom, all over the bottom squared"
Chain rule - Answers "Outside-Inside" Rule
d/dx sin(x) - Answers cos(x)
d/dx cos(x) - Answers -sin(x)
d/dx tan(x) - Answers sec²(x)
d/dx sec(x) - Answers sec(x)tan(x)
d/dx csc(x) - Answers -csc(x)cot(x)
d/dx cot(x) - Answers -csc²(x)
d/dx arctan(x) - Answers 1 / (1+x²)
d/dx ln(x) - Answers 1 / x
d/dx e^x - Answers e^x
Velocity - Answers Derivative of position
Acceleration - Answers Derivative of velocity
