WGU C958
CALCULUS I
Comprehensive Study Guide
Concepts - Formulas - Graph Interpretation - Worked Examples - Practice Checks
2026 Independent Study Edition
A structured review of functions, limits, derivatives, applications of derivatives, antiderivatives, definite integrals,
and the Fundamental Theorem of Calculus.
Educational use: This guide contains original explanations and practice. It is not an official WGU publication and does not reproduce an
Objective Assessment.
,WGU C958 Calculus I - Comprehensive Study Guide
How to Use This Guide
Calculus is cumulative: weaknesses in algebra or functions become derivative and integral errors later. Use the diagnostic
first, study in order, and solve examples without looking at the worked steps.
Recommended four-pass method
• Pass 1 - Diagnose: complete the readiness check and mark uncertain skills.
• Pass 2 - Learn: read each concept and recreate the worked examples.
• Pass 3 - Retrieve: answer every checkpoint without notes.
• Pass 4 - Integrate: complete the mixed review, then explain each error in words.
Calculator discipline
Use a calculator to verify arithmetic and visualize graphs, not to replace setup. Always identify the function,
operation, units, and relevant theorem before entering values.
Core competency map
Concept Formula / condition Interpretation
Functions domain, range, composition, inverse Read and transform mathematical models.
Limits lim f(x) as x approaches a Describe local behavior and continuity.
Derivatives f'(x) = lim[h->0] (f(x+h)-f(x))/h Measure instantaneous rate of change.
Applications critical points: f'(x)=0 or DNE Optimize and analyze graphs.
Integrals antiderivative and signed accumulation Reverse differentiation and total change.
FTC integral[a,b] f(x)dx = F(b)-F(a) Connect accumulation to derivatives.
Diagnostic Readiness Check
Attempt without notes. Answers appear immediately below so you can build a targeted plan.
1. Simplify (x^2 - 9)/(x - 3). 2. Solve e^(2x)=7. 3. State the domain of 1/(x-4). 4. Find the slope through (2,5) and (6,13).
5. Evaluate sin(pi/6). 6. Rewrite ln(ab^2). 7. If f(x)=x^2+1 and g(x)=3x, find f(g(2)). 8. Factor x^2-5x+6.
Diagnostic answers
1. x+3 for x != 3. 2. x = ln(7)/2. 3. All real x except 4. 4. 2. 5. 1/2. 6. ln(a)+2ln(b), for positive a and b. 7. 37. 8.
(x-2)(x-3).
Prerequisite repair
• Fractions: use a common denominator before combining; cancellation only removes factors, not terms.
• Exponents: x^a x^b=x^(a+b); (x^a)^b=x^(ab); x^(-a)=1/x^a.
• Logs: ln(ab)=ln a+ln b; ln(a/b)=ln a-ln b; ln(a^r)=r ln a.
• Trig: know the unit-circle values at 0, pi/6, pi/4, pi/3, and pi/2.
• Equations: isolate the transformed expression before applying inverse operations.
Original independent study resource - not affiliated with or endorsed by WGU Page 2
, WGU C958 Calculus I - Comprehensive Study Guide
1. Functions, Models, and Graphs
A function assigns exactly one output to each allowed input. Calculus statements are only meaningful when the function,
its domain, and the variable are clear.
Function language
Concept Formula / condition Interpretation
Domain allowable x-values Exclude zero denominators, even roots of negatives,
and nonpositive log inputs.
Range attainable y-values Read from outputs or solve y=f(x) for x.
Composition (f o g)(x)=f(g(x)) Apply the inside function first.
Inverse f^(-1)(f(x))=x Swap x and y, solve for y; restrict domain if needed.
Average rate [f(b)-f(a)]/(b-a) Slope of the secant line.
Worked example: Domain and average rate
Step 1. For f(x)=sqrt(x-1)/(x-5), require x-1 >= 0, so x >= 1.
Step 2. Also require x != 5. Thus the domain is [1,5) union (5,infinity).
Step 3. For g(t)=t^2-3t, average rate on [1,4] is [g(4)-g(1)]/(4-1).
Step 4. g(4)=4 and g(1)=-2, so the rate is 6/3.
Answer
Domain: [1,5) union (5,infinity). Average rate: 2 units of output per unit of input.
Transformations
• y=f(x)+k shifts vertically by k; y=f(x-h) shifts right by h.
• y=af(x) scales vertically by |a| and reflects across the x-axis if a<0.
• y=f(bx) scales horizontally by factor 1/|b| and reflects across the y-axis if b<0.
Common trap
For horizontal shifts, the sign appears reversed: f(x-3) moves right 3, while f(x+3) moves left 3.
Checkpoint 1
A. Find the domain of ln(x-2). B. If f(x)=2x+1, find f^(-1)(x). C. Compute the average rate of x^3 from x=1 to x=3.
Answers
A. (2,infinity). B. (x-1)/2. C. (27-1)/(3-1)=13.
2. Limits and Continuity
A limit describes what f(x) approaches as x approaches a value. It may exist even when f(a) is missing or different.
Original independent study resource - not affiliated with or endorsed by WGU Page 3
CALCULUS I
Comprehensive Study Guide
Concepts - Formulas - Graph Interpretation - Worked Examples - Practice Checks
2026 Independent Study Edition
A structured review of functions, limits, derivatives, applications of derivatives, antiderivatives, definite integrals,
and the Fundamental Theorem of Calculus.
Educational use: This guide contains original explanations and practice. It is not an official WGU publication and does not reproduce an
Objective Assessment.
,WGU C958 Calculus I - Comprehensive Study Guide
How to Use This Guide
Calculus is cumulative: weaknesses in algebra or functions become derivative and integral errors later. Use the diagnostic
first, study in order, and solve examples without looking at the worked steps.
Recommended four-pass method
• Pass 1 - Diagnose: complete the readiness check and mark uncertain skills.
• Pass 2 - Learn: read each concept and recreate the worked examples.
• Pass 3 - Retrieve: answer every checkpoint without notes.
• Pass 4 - Integrate: complete the mixed review, then explain each error in words.
Calculator discipline
Use a calculator to verify arithmetic and visualize graphs, not to replace setup. Always identify the function,
operation, units, and relevant theorem before entering values.
Core competency map
Concept Formula / condition Interpretation
Functions domain, range, composition, inverse Read and transform mathematical models.
Limits lim f(x) as x approaches a Describe local behavior and continuity.
Derivatives f'(x) = lim[h->0] (f(x+h)-f(x))/h Measure instantaneous rate of change.
Applications critical points: f'(x)=0 or DNE Optimize and analyze graphs.
Integrals antiderivative and signed accumulation Reverse differentiation and total change.
FTC integral[a,b] f(x)dx = F(b)-F(a) Connect accumulation to derivatives.
Diagnostic Readiness Check
Attempt without notes. Answers appear immediately below so you can build a targeted plan.
1. Simplify (x^2 - 9)/(x - 3). 2. Solve e^(2x)=7. 3. State the domain of 1/(x-4). 4. Find the slope through (2,5) and (6,13).
5. Evaluate sin(pi/6). 6. Rewrite ln(ab^2). 7. If f(x)=x^2+1 and g(x)=3x, find f(g(2)). 8. Factor x^2-5x+6.
Diagnostic answers
1. x+3 for x != 3. 2. x = ln(7)/2. 3. All real x except 4. 4. 2. 5. 1/2. 6. ln(a)+2ln(b), for positive a and b. 7. 37. 8.
(x-2)(x-3).
Prerequisite repair
• Fractions: use a common denominator before combining; cancellation only removes factors, not terms.
• Exponents: x^a x^b=x^(a+b); (x^a)^b=x^(ab); x^(-a)=1/x^a.
• Logs: ln(ab)=ln a+ln b; ln(a/b)=ln a-ln b; ln(a^r)=r ln a.
• Trig: know the unit-circle values at 0, pi/6, pi/4, pi/3, and pi/2.
• Equations: isolate the transformed expression before applying inverse operations.
Original independent study resource - not affiliated with or endorsed by WGU Page 2
, WGU C958 Calculus I - Comprehensive Study Guide
1. Functions, Models, and Graphs
A function assigns exactly one output to each allowed input. Calculus statements are only meaningful when the function,
its domain, and the variable are clear.
Function language
Concept Formula / condition Interpretation
Domain allowable x-values Exclude zero denominators, even roots of negatives,
and nonpositive log inputs.
Range attainable y-values Read from outputs or solve y=f(x) for x.
Composition (f o g)(x)=f(g(x)) Apply the inside function first.
Inverse f^(-1)(f(x))=x Swap x and y, solve for y; restrict domain if needed.
Average rate [f(b)-f(a)]/(b-a) Slope of the secant line.
Worked example: Domain and average rate
Step 1. For f(x)=sqrt(x-1)/(x-5), require x-1 >= 0, so x >= 1.
Step 2. Also require x != 5. Thus the domain is [1,5) union (5,infinity).
Step 3. For g(t)=t^2-3t, average rate on [1,4] is [g(4)-g(1)]/(4-1).
Step 4. g(4)=4 and g(1)=-2, so the rate is 6/3.
Answer
Domain: [1,5) union (5,infinity). Average rate: 2 units of output per unit of input.
Transformations
• y=f(x)+k shifts vertically by k; y=f(x-h) shifts right by h.
• y=af(x) scales vertically by |a| and reflects across the x-axis if a<0.
• y=f(bx) scales horizontally by factor 1/|b| and reflects across the y-axis if b<0.
Common trap
For horizontal shifts, the sign appears reversed: f(x-3) moves right 3, while f(x+3) moves left 3.
Checkpoint 1
A. Find the domain of ln(x-2). B. If f(x)=2x+1, find f^(-1)(x). C. Compute the average rate of x^3 from x=1 to x=3.
Answers
A. (2,infinity). B. (x-1)/2. C. (27-1)/(3-1)=13.
2. Limits and Continuity
A limit describes what f(x) approaches as x approaches a value. It may exist even when f(a) is missing or different.
Original independent study resource - not affiliated with or endorsed by WGU Page 3
