Calculus 1000A
Summer 2023
Multiple Choice Questions
Practice Midterm
1. Thefunctionf(x)=loga(x+1)+bwhosegraphpassesthroughpointsP(0,2)andQ(8,0)is? Answer: f(x) = log1/3(x + 1) + 2.
2. The function f (x) = 4 − abx whose graph passes through the point P (2, 0) is? Answer: f(x) = 4 − 2x.
3. Find the function f−1(x), if f(x) = 3(lnx)5. √
Answer: f−1(x) = e 5 x/3.
4. Find the function f−1(x), if f(x) = cos(ln(√x)).
Answer: f−1(x) = e2 arccos x.
5. Find the domain of the function f(x) = sin−1(e2x).
Answer: (−∞,0].
6. Find the domain of the function f(x) = ln(sin−1(x)).
Answer: (0,1].
7. Evaluate tan(arcsin(45)).
Answer: 34.
8. Evaluate arcsin(cos(− 3π )).
Answer: −π4.
9. What inequalities are satisfied by the numbers x, y, z, if
x = log3(log2 8), y = 3arccos(−1), z = ln4 ? ln2
Answer: x
Calculus 1000A Summer 2023
29. If f(x) = cos(x + π2 ) and g(x) is a function such that g′′(0) = 2023 and (fg)′′(0) = 4, then g′(0) = ?
Answer: −2.
30. If f(x) = sin(2x), g(x) = cos(4x), and h(x) is a function such that h′(π/4) = 3, then
(fgh)′(π/4) = ? Answer: −3.
Short Answer Questions
1. Use the Squeeze Theorem to find lim (e−1/x · cos( 2023 )). Justify your answer. x→0+ x
2. Use the Squeeze Theorem to find lim (sin( πx ) · ln(cos2 x + 1)). Justify your answer. x→∞
3. Use the Intermediate Value Theorem to show that the equation tan(x − π6 ) = 2cos x
has a solution. Justify your answer.
4. Use the Intermediate Value Theorem to show that the equation
x3 +2x+1=e−x +5
has a solution. Justify your answer.
5. Let A and B be real numbers and let
Aarcsin(esinx), x<0
f(x)= B, x=0 √
2− x+4, x>0. x
(a) Find lim f(x) and lim f(x).
(b) For what values of A and B is the function f continuous at 0? Justify your answer.
6. Let f(x) = logC(x2). Find the value of C for which the tangent line to the graph of f(x) at the point P (1, 0) coincides with the tangent line to the curve (x − 2)2 + (y − 1)2 = 2 at the same point. Justify your answer.
浙大学霸代写 加微信 cstutorcs