به انگلیسی

Exponentials wearing trig costumes

Mathematics · Algebra · ages 17-18

Name ______________________ Date ____________

What is 4 sinh(ln 2)? Give an exact number.
Answer: ______________
For every real a, which pair correctly describes the symmetry of sinh and cosh?
- sinh(-a) = -sinh a; cosh(-a) = cosh a
- sinh(-a) = sinh a; cosh(-a) = -cosh a
- sinh(-a) = -sinh a; cosh(-a) = -cosh a
- sinh(-a) = sinh a; cosh(-a) = cosh a
Which defines sinh x in terms of exponentials?
- (e to the x plus e to the minus x) over 2
- (e to the x minus e to the minus x) over 2
- e to the x minus e to the minus x
Zoe says cosh squared x minus sinh squared x equals 1 for every real x. Is Zoe right?
Circle one: True False
Let h(x) = arsinh x. What is h'(sqrt(15))? Give a decimal.
Answer: ______________
For a real x, sinh x = 3. What is cosh squared x?
- 10
- 12
- 18
As x moves from -2 through 0 to 2, how does cosh x change?
- It increases to 1, then decreases.
- It decreases to 1, then increases.
- It decreases to 0, then increases.
- It increases to 0, then decreases.
Using the principal real inverse, arcosh(cosh(-4)) = -4. True or false?
Circle one: True False
Max says arsinh x, which equals ln(x plus root(x squared plus 1)), is defined for every real x. Is Max right?
Circle one: True False
Which logarithmic formula and domain describe the principal real arcosh x?
- ln(x + sqrt(x squared + 1)), x >= 1
- ln(x + sqrt(x squared - 1)), x <= -1
- ln(x - sqrt(x squared - 1)), x >= 1
- ln(x + sqrt(x squared - 1)), x >= 1

Answer key

For grown-ups. Fold this page away before handing over the rest.

3 · The exponential values are 2 and 1/2, so 4 sinh(ln 2) = 4(2 - 1/2)/2 = 3.
sinh(-a) = -sinh a; cosh(-a) = cosh a · Sinh is odd, so its value changes sign. Cosh is even, so its value stays the same.
(e to the x minus e to the minus x) over 2 · sinh x is the half difference, while the half sum defines cosh x.
True · The pure exponential square terms cancel. Subtracting the cross terms leaves 4 in the numerator, so the difference is 4/4 = 1.
0.25 · Use h'(x) = 1/sqrt(1 + x squared). At sqrt(15), this becomes 1/sqrt(16) = 1/4 = 0.25.
10 · The identity gives cosh squared x = 1 + sinh squared x = 1 + 9 = 10.
It decreases to 1, then increases. · Cosh decreases on negative inputs, reaches its minimum 1 at zero, then increases on positive inputs.
False · Cosh(-4) equals cosh 4. The principal arcosh selects the nonnegative input 4.
True · Max is right. The log input stays positive for all real x.
ln(x + sqrt(x squared - 1)), x >= 1 · The principal formula is ln(x + sqrt(x squared - 1)), for x >= 1. It returns nonnegative outputs.