Score: Name:
Sample Final Exam Calculus 1 - MAT 175
Instructions:This exam should be taken without text or notes or electronic devices. Show your work, and indicate answers
clearly. Cross out all work that you do not want to be graded.
1. (5pts.) Compute the derivative dy
dx for y=√5 + x5
5−5 ln x.
2. (5pts.) Compute the derivative p0(y) of the function p(y) = 2y+3
7y+5 .
3. (5pts.) Write down an equation of the tangent line to the graph of y= 1 + 2exat the point where x= 0.
4. (5pts.) Determine the slope of the tangent line to the graph of the equation 5x2+ 3y2+xy = 15 at the point
(−1,2).
5. (5pts.) Compute the derivative dw
dz of the function w=√1 + zez.
6. (5pts.) Find the limit:
lim
θ→0
3 sin 4θ
5θ=
7. (5pts.) Find the limit:
lim
x→1
x2−4x+ 3
x2−1=
8. (5pts.) Find the limit:
lim
x→∞
6x2+ 100
7x2−100 =
9. (5pts.) For which constant kis the following function Q(x) continuous for all x? Justify your answer.
Q(x) = (x2+kif x≤0
cos xif 0 < x
10. (6pts.) Show that the derivative of f(x) = 1
xis f0(x) = −1
x2by using the definition of the derivative as the limit
of a difference quotient.
11. (5pts.) If the volume V=s3of an expanding cube is increasing at the constant rate of 120 cubic inches per
second, how fast is the length sof the sides increasing when the volume is 8 cubic inches?
12. (5pts.) Find where the graph of y=x3−6x2is concave up and concave down, and find all inflection points.
13. (5pts.) Find and classify all the relative extrema of F(x) = x4−4x2+ 2.
14. (6pts.) Find the absolute maximum and minimum values of f(x)=4x3−3x2on the closed interval [−1,1].
15. (6pts.) A particle moves along the x-axis with an acceleration given by a(t) = 6t+ 2, where tis measured
in seconds and s(position) is measured in meters. If the initial position is given by s(0) = 3 and the initial
velocity is given by v(0) = 4 then find the position of the particle at tseconds.
16. (5pts.) A rectangular poster is to have an area of 100 square inches with a 1-inch margin on the right and left
sides and a 2-inch margin at the top and bottom. Find the dimensions of the poster with the largest printed
area.
