#1
Which of the following is the exponential function?
f(x) = e^x
ExplanationExponential function with base e.
#2
If p(x) = 5 * e^x, what is the value of p(0)?
5
ExplanationEvaluate the function at x=0.
#3
Simplify the expression log(e^3).
3
ExplanationLogarithm of base e.
#4
If h(x) = 5 * e^(2x), what is the value of h(1)?
25
ExplanationEvaluate the function at x=1.
#5
What is the solution to the equation 2^x = 8?
x = 3
ExplanationExponentiation function solving for x.
#6
If f(x) = 3 * 2^x, what is the value of f(2)?
18
ExplanationSubstitute x=2 into the exponential function.
#7
What is the derivative of the function f(x) = e^(2x)?
2e^(2x)
ExplanationApply the chain rule for differentiation.
#8
If q(x) = 2 * e^(x-1), what is the value of q(2)?
6e
ExplanationSubstitute x=2 into the exponential function.
#9
If r(x) = 4 * e^(2x), what is the value of r(-1)?
4
ExplanationEvaluate the function at x=-1.
#10
What is the value of 2^(log2(5))?
5
ExplanationLogarithmic property simplification.
#11
If g(x) = e^(2x) + 3e^x + 2, what is the minimum value of g(x)?
2
ExplanationDetermine where the derivative is zero.
#12
Solve the equation log(2x) = 3.
x = 2
ExplanationSolve for x using logarithmic properties.
#13
If f(x) = 2^x - 3, what is the x-intercept of the graph?
2
ExplanationFind where the function equals zero.
#14
If p(x) = e^x + 2, what is the range of p(x)?
Only positive real numbers
ExplanationExponential function output.
#15
What is the domain of the function g(x) = e^x?
All real numbers
ExplanationThe function is defined for all real numbers.
#16
If h(x) = 4^(x-1), what is the value of h(3)?
64
ExplanationEvaluate the function for x=3.
#17
Solve the equation 3^(2x+1) = 27.
x = 1
ExplanationFind the value of x satisfying the equation.
#18
What is the limit of f(x) = (1 + e^x)/(1 - e^x) as x approaches infinity?
1
ExplanationAnalyze the behavior of the function as x grows.
#19
What is the general solution to the differential equation dy/dx = 2e^(2x)?
y = Ce^(2x)
ExplanationSolution incorporating the constant of integration.
#20
If f(x) = e^(3x) - e^(2x), what is the critical point?
x = 1
ExplanationDerivative equals zero for critical point.
#21
Evaluate the limit lim(x->0) [(e^x - 1) / x].
e
ExplanationL'Hôpital's Rule application.
#22
What is the inverse of the function y = ln(x)?
y = e^x
ExplanationSwitch x and y then solve for y.
#23
Solve the equation 4^(x-1) = 8^x.
x = 2
ExplanationEqual bases, equate exponents.
#24
What is the derivative of the function g(x) = ln(3x)?
1/x
ExplanationDerivative using logarithmic differentiation.
#25
Evaluate the limit lim(x->infinity) (1 + 2e^x)/(3 - e^x).
2
ExplanationAnalyze the behavior of the function as x grows.